Artykuły w czasopismach na temat „Odd colouring”

We consider distance colourings in graphs of maximum degree at most d and how excluding one fixed cycle of length ℓ affects the number of colours required as d → ∞. For vertex-colouring and t ⩾ 1, if any two distinct vertices connected by a path of at most t edges are required to be coloured differently, then a reduction by a logarithmic (in d) factor against the trivial bound O(dt) can be obtained by excluding an odd cycle length ℓ ⩾ 3t if t is odd or by excluding an even cycle length ℓ ⩾ 2t + 2. For edge-colouring and t ⩾ 2, if any two distinct edges connected by a path of fewer than t edges are required to be coloured differently, then excluding an even cycle length ℓ ⩾ 2t is sufficient for a logarithmic factor reduction. For t ⩾ 2, neither of the above statements are possible for other parity combinations of ℓ and t. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).

AbstractAn $r$-ary necklace (bracelet) of length $n$ is an equivalence class of $r$-colourings of vertices of a regular $n$-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric $r$-ary necklaces (bracelets) of length $n$ is $\frac{1}{2} (r+ 1){r}^{n/ 2} $ if $n$ is even, and ${r}^{(n+ 1)/ 2} $ if $n$ is odd.

AbstractAs a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no oddKtminor is (t− 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for eacht⩾ 2, every graph with no oddKtminor has a partition of its vertex set into 6t− 9 setsV1, …,V6t−9such that eachViinduces a subgraph of bounded maximum degree. Secondly, we prove that for eacht⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t−13 setsV1,…,V10t−13such that eachViinduces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into 496tsuch sets.

Gerards and Seymour (see [10], p. 115) conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colourable. This is an analogue of the well-known conjecture of Hadwiger, and in fact, this would immediately imply Hadwiger's conjecture. The current best-known bound for the chromatic number of graphs with no odd complete minor of order l is $O(l \sqrt{\log l})$ by the recent result by Geelen, Gerards, Reed, Seymour and Vetta [8], and by Kawarabayashi [12] later, independently. But it seems very hard to improve this bound since this would also improve the current best-known bound for the chromatic number of graphs with no complete minor of order l.Motivated by this problem, in this note we show that there exists an absolute constant f(k) such that any graph G with no odd complete minor of order k admits a vertex partition V1, . . ., V496k such that each component in the subgraph induced on Vi (i ≥ 1) has at most f(k) vertices. When f(k) = 1, this is a colouring of G. Hence this is a relaxation of colouring in a sense, and this is the first result in this direction for the odd Hadwiger's conjecture.Our proof is based on a recent decomposition theorem due to Geelen, Gerards, Reed, Seymour and Vetta [8], together with a connectivity result that forces a huge complete bipartite minor in large graphs by Böhme, Kawarabayashi, Maharry and Mohar [3].

For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

AbstractWe find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n−1)-edge colouring of Kn (n even), and for n-edge colouring of Kn (n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.

Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

Claude Shannon’s 1948 paper ‘A Mathematical Theory of Communication’ provided the essential foundation for the digital/information revolution that enables these very pixels to glow in meaningful patterns and permeates nearly every aspect of modern life. Information Theory, born fully grown from this paper, has been applied and mis-applied to a multitude of disciplines in the last 70-odd years, from quantum physics to psychology. Shannon himself famously decried those jumping on the ‘scientific bandwagon’ of Information Theory without sufficient mathematical rigour. Nevertheless, having a brief personal connection to Dr Shannon (and being extremely grateful for it), I will take the liberty of colouring some of my experience with computer network music with less-than-rigorous insights gained from his work.

We say that Kn → (G,H), if for every red/blue colouring of edges of the complete graph Kn, there exists a red copy of G, or a blue copy of H in the colouring of Kn. The Ramsey number r(G,H) is the smallest positive integer n such that Kn → (G,H). Let r(n,m)=r(Kn, Km). A closely related concept of Ramsey numbers is the Star-critical Ramsey number r*(G, H) defined as the largest value of k such that K r(G,H)-1 ˅ K 1,k → (G,H). Literature on survey papers in this area reveals many unsolved problems related to these numbers. One of these problems is the calculation of Ramsey numbers for certain classes of graphs. The primary objective of this paper is to calculate the Star critical Ramsey numbers for the case of Stars versus K1,m+e. The methodology that we follow in solving this problem is to first find a closed form for the Ramsey number r*(K1,n , K1,m+e) for all n, m ≥ 3. Based on the values of r*(K1,n , K1,m+e) for different n, m we arrive at a general formula for r*(K1,n , K1,m+e). Henceforth, we show that r*(K1,n , K1,m+e) = n+m-1 is defined by a piecewise function related to the three disjoint cases of n, m both even and n ≤ m - 2, n or m is odd and n ≤ m-2 and n > m-2.

We consider versions of the grasshopper problem (Goulko & Kent 2017 Proc. R. Soc. A 473 , 20170494) on the circle and the sphere, which are relevant to Bell inequalities. For a circle of circumference 2 π , we show that for unconstrained lawns of any length and arbitrary jump lengths, the supremum of the probability for the grasshopper’s jump to stay on the lawn is one. For antipodal lawns, which by definition contain precisely one of each pair of opposite points and have length π , we show this is true except when the jump length ϕ is of the form π ( p / q ) with p , q coprime and p odd. For these jump lengths, we show the optimal probability is 1 − 1/ q and construct optimal lawns. For a pair of antipodal lawns, we show that the optimal probability of jumping from one onto the other is 1 − 1/ q for p , q coprime, p odd and q even, and one in all other cases. For an antipodal lawn on the sphere, it is known (Kent & Pitalúa-García 2014 Phys. Rev. A 90 , 062124) that if ϕ = π / q , where q ∈ N , then the optimal retention probability of 1 − 1/ q for the grasshopper’s jump is provided by a hemispherical lawn. We show that in all other cases where 0 < ϕ < π /2, hemispherical lawns are not optimal, disproving the hemispherical colouring maximality hypotheses (Kent & Pitalúa-García 2014 Phys. Rev. A 90 , 062124). We discuss the implications for Bell experiments and related cryptographic tests.

AbstractFor given graphs G1,…, Gk, the size-Ramsey number $\hat R({G_1}, \ldots ,{G_k})$ is the smallest integer m for which there exists a graph H on m edges such that in every k-edge colouring of H with colours 1,…,k, H contains a monochromatic copy of Gi of colour i for some 1 ≤ i ≤ k. We denote $\hat R({G_1}, \ldots ,{G_k})$ by ${\hat R_k}(G)$ when G1 = ⋯ = Gk = G.Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n, ${\hat R_k}({C_n}) \le {c_k}n$ for some constant ck. Their proof, however, is based on Szemerédi’s regularity lemma so no specific constant ck is known.In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of ${\hat R_k}({C_n}) \le {c_k}n$ , avoiding use of the regularity lemma, where ck is exponential and doubly exponential in k, when n is even and odd, respectively. In particular, we show that for sufficiently large n we have ${\hat R_2}({C_n}) \le {10^5} \times cn$ , where c = 6.5 if n is even and c = 1989 otherwise.

AbstractGiven graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem.We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4).To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

Objective: The current study focuses on R-Phycoerythrin pigment production from Seaweed using different chemical and physical conditions. Methods: In the present study Seaweed was collected from Rameshwaram and identified by CS-MCRI Institute, Mandapam. The collected seaweed was then washed using distilled water for further processing. Using a sterile knife the seaweed was cut into small pieces. The chopped seaweeds were then weighed and subjected to different optimization procedures for pigment production. These equally weighed seaweeds were treated with three varying Buffers at different pH, the buffer showing better O.D value was subjected to different Cell disruption techniques and finally freeze thawed at different temperature stress.Results: The seaweeds were subjected to different chemical and physical stress conditions for R-phycoerythrin production. On optimizing the different buffer solutions for pigment production Sodium phosphate buffer showed maximum O.D of 0.215 when compared to other buffers whereas on providing different pH conditions the O.D value obtained was high at pH 7.2. Different cell disruption techniques were followed for pigment production using the sodium phosphate buffer at pH 7.2 and freeze thaw method was found suitable for the highest pigment production with O.D value of 0.441. Hence after optimization of different extraction procedures, cell disruption followed by freeze & thaw method (−20°C and 25°C) showed maximum R-phycoerythrin content. Conclusion: From the findings, it was also observed that the primary metabolites produced by these organisms may serve as potential bioactive compounds of interest in the Food industries as natural colourant and in cosmetic industries.Keywords: Seaweeds, Extraction, Phycoerythrin, Optimization, Cell disruption, Sonication.

Background: Skin bleaching was reported to be commonly practiced among women and Africa was reported to be one of the most affected yet the subject is not given much attention in public health research in Zimbabwe despite the adverse effects of skin bleaching on health. Method: This study was an exploratory cross-sectional survey to explore skin bleaching, skin bleaching patterns and factors associated with skin bleaching among women living in Zimbabwe. An online self-administered questionnaire was sent out to women on social network i.e. WhatsApp, Facebook, LinkedIn and Twitter. Findings: A total number of 260 respondents, mean age 31.69 (SD, 8.12) years participated in the survey. The prevalence of skin bleaching among the participants was 31.15%. The major reason reported for skin bleaching was to have smooth and healthy skin alongside other factors such as beauty, gaining social favours for example getting married and good jobs. Occupation, complexion and marital status were associated with skin bleaching. The odds of skin bleaching for participants who were employed was 1.45(95% confidence interval [CI],0.32-1.91);p-value 0.02, dark skinned participants 2.56(95% CI, 0.76-2.87);p-value 0.01 and unmarried participants 2.87(95% CI,0.29-3.58);p-value 0.03. Conclusion: Evidence from the research suggests skin bleaching might be common among women living in Zimbabwe and possibly poses serious health threats to the women. Skin bleaching seems to be deep rooted in colourism. The colourism seems to be taken advantage of by the cosmetic industry which produce the potentially hazardous products which promise the revered light skin to women but which comes with a price. However, the study provides a base for future studies to ex- plore more on skin bleaching practices among women living in Zimbabwe. Keywords: Skin bleaching; skin bleaching products; women; Zimbabwe.

AbstractPetruševski and Škrekovski recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph G is said to be odd if for each non-isolated vertex $$x \in V(G)$$ x ∈ V ( G ) there exists a colour c appearing an odd number of times in its neighbourhood N(x). Petruševski and Škrekovski proved that for any planar graph G there is an odd colouring using at most 9 colours and, together with Caro, showed that 8 colours are enough for a significant family of planar graphs. We show that 8 colours suffice for all planar graphs.

The Hales–Jewett Theorem states that any $r$–colouring of $[m]^n$ contains a monochromatic combinatorial line if $n$ is large enough. Shelah's proof of the theorem implies that for $m = 3$ there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most $r$ intervals. For odd $r$, Conlon and Kamčev constructed $r$–colourings for which it cannot be fewer than $r$ intervals. However, we show that for even $r$ and large $n$, any $r$–colouring of $[3]^n$ contains a monochromatic combinatorial line whose set of active coordinates is the union of at most $r-1$ intervals. This is optimal and extends a result of Leader and Räty for $r=2$.

AbstractAn ‐cycle decomposition of is said to be equitably 2‐colourable if there is a 2‐vertex‐colouring of such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle of the decomposition, each colour appears on or of the vertices of . In this paper we study the existence of equitably 2‐colourable ‐cycle decompositions of , where is odd, and prove the existence of such a decomposition for (mod ).

Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$, for any $v \in V(G)$. We show that every digraph $G$ equipped with an assignment of lists $L$, each of size at least $k$, has a $2/k$-majority $L$-colouring. For even $k$ this is best possible, while for odd $k$ the constant $2/k$ cannot be replaced by any number less than $2/(k+1)$. This generalizes a result of Anholcer, Bosek and Grytczuk, who proved the cases $k=3$ and $k=4$ and claim a weaker result for general $k$.

International audience The colouring number col($G$) of a graph $G$ is the smallest integer $k$ for which there is an ordering of the vertices of $G$ such that when removing the vertices of $G$ in the specified order no vertex of degree more than $k-1$ in the remaining graph is removed at any step. An edge $e$ of a graph $G$ is said to be <i>double</i>-col-<i>critical</i> if the colouring number of $G-V(e)$ is at most the colouring number of $G$ minus 2. A connected graph G is said to be double-col-critical if each edge of $G$ is double-col-critical. We characterise the <i>double</i>-col-<i>critical</i> graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer $k$ greater than 4 and any positive number $&epsilon;$, there is a $k$-col-critical graph with the ratio of double-col-critical edges between $1- &epsilon;$ and 1.

We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every odd integer $d$, every multigraph $G$ with maximum degree $\Delta$ is $(\lceil \frac{3\Delta - 1}{3d - 1} \rceil, d)$-edge colourable and this bound is attained for all values of $\Delta$ and $d$. An easy consequence of Vizing's Theorem is that, for every (simple) graph $G,$ $\chi'_{d}(G) \in \{ \lceil \frac{\Delta}{d} \rceil, \lceil \frac{\Delta+1}{d} \rceil \}$. We characterize the values of $d$ and $\Delta$ for which it is NP-complete to compute $\chi'_d(G)$. These results generalize classic results on the chromatic index of a graph by Shannon, Holyer, Leven and Galil and extend a result of Amini, Esperet and van den Heuvel.

Abstract Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$ -minor admits a vertex $(2t-2)$ -colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ . The bound on the number of colours is optimal up to a factor of $2$ , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput.17 815–821), Kang and Oum (2019, Combin. Probab. Comput.28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc.98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$ -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$ -minor-free graphs with some additional ideas, which make the extension to odd $K_t$ -minor-free graphs possible.

International audience A strong stable set in a graph $G$ is a stable set that contains a vertex of every maximal clique of $G$. A Meyniel obstruction is an odd circuit with at least five vertices and at most one chord. Given a graph $G$ and a vertex $v$ of $G$, we give a polytime algorithm to find either a strong stable set containing $v$ or a Meyniel obstruction in $G$. This can then be used to find in any graph, a clique and colouring of the same size or a Meyniel obstruction.

Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic index of $G$ is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than $t$ edges must receive different colours. Let $\pi'_t(d)$ and $\tau'_t(d)$ be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree $d$. We have that $\pi'_t(d)$ is at most and at least a non-trivial constant multiple larger than $\tau'_t(d)$. (We conjecture $\limsup_{d\to\infty}\pi'_2(d)/\tau'_2(d) =9/4$ in particular.) We prove for odd $t$ the existence of a quantity $g$ depending only on $t$ such that the distance-$t$ chromatic index of any planar multigraph of maximum degree $d$ and girth at least $g$ is at most $\tau'_t(d)$ if $d$ is sufficiently large. Such a quantity does not exist for even $t$. We also show a related, similar phenomenon for distance vertex-colouring.

Abstract. An r-graph is an r-regular graph G on an even number of vertices where every odd set X ⊆V(G) is connected by at least r edges to its complement V(G) \ X. Every r-graph has a perfect matching and in a poorly matchable r-graph every pair of perfect matchings intersect, which implies that poorly matchable r-graphs are not r-edge-colourable. We prove, for each fixed r ≥ 3, that poorly matchable r-graph recognition is coNP-complete, an indication that, for each odd d ≥ 3, it may be a hard problem to recognise d-regular (d−1)-edge-connected non-d-edge-colourable graphs, referred to as d-snarks in this paper. We show how to construct, for every fixed odd d ≥ 5, an infinite family of d-snarks. These families provide a natural extension to the well-known Loupekine snarks. We also discuss how the hunting of the smallest d-snarks may help in strengthening and better understanding the major Overfull Conjecture on edge-colouring simple graphs.

Vertex colouring is a much studied problem in combinatorics and computer science for its theoretical as well as its practical aspects. In this paper<br />we are concerned with the "distributed" version of a question stated by Vizing, concerning a Brooks-like theorem for sparse graphs. Roughly, the question asks whether there exist colourings using many fewer than<br />Delta colours, where Delta denotes the maximum degree of the graph, provided that some sparsity conditions are satisfied. In this paper we show that such colourings not only exist, but that they can be quickly computed by extremely simple distributed, randomized algorithms. Before stating our results precisely, we review the relevant facts. For any graph G of maximum degree with n vertices, the following trivial algorithm computes a Delta+1 (list) colouring. Each vertex u initially has a list, or palette, of deg(u) + 1 colours. The computation proceeds<br />in rounds. During each round, each uncoloured vertex, in parallel, first performs a trivial attempt: it picks a tentative colour at random from its palette and if no neighbour picks the same colour, the colour becomes<br />final and the algorithm stops for that vertex. Otherwise, the vertex's palette undergoes a trivial update - the colours succesfully used by the neigbours are removed - and a new attempt is performed in the next round.<br />Henceforth we shall refer to this as "the" trivial algorithm. The trivial algorithm always computes a valid colouring regardless of the composition<br />of the initial lists, and does so in O(log n) rounds with high<br />probability|that is, with probability approaching 1 as the number of vertices increases [10, 13, 4].<br />It is apparent that the trivial algorithm is distributed, since each<br />vertex only relies on information from the neighbouring vertices. The well-known distributed algorithm for the same problem given by Luby [15] amends the trivial algorithm in the following way: at the beginning of each round every uncoloured vertex is asleep. Each such vertex awakes<br />with probability p and executes a trivial attempt (in Luby's paper p = 1/2). Then, whether or not the vertex awoke, the palette undergoes a trivial update. At the end of the round the vertex goes back to sleep. <br /> <br /> We shall refer to this variant of the trivial algorithm as the dozing-off algorithm. The dozing-off algorithm has the same asymptotic performance<br />as the trivial algorithm, but its analysis just needs pairwise independence.<br />Luby used this fact to carry out a derandomization procedure<br />in the pram model. Can better colourings|i.e. colourings using fewer colours|be computed<br />eciently in a distributed setting? In 1948 Brooks gave a theorem<br />that characterizes the graphs which are not {colourable: a graph is<br />{colourable if and only if it is neither an odd cycle nor a + 1 clique (see, for instance, [2]).<br />The proof of Brooks' theorem is actually a polynomial time sequential algorithm. {colourings can also be quickly (i.e. in polylogarithmic time) computed in the PRAM model [8, 11, 12]. In fact, a distributed<br />version of Brooks' theorem can be derived from a certain locality property of -colourings, yielding the following: There is no o(n) randomized, synchronous protocol to -colour paths, cycles or cliques. For all other graphs, there is a randomized protocol which, with high probability, computes a -colouring in polylogarithmically many rounds [17].<br />(The property in question, holding for graphs which are neither cliques, paths nor cycles, is this: If G is {coloured except for one last vertex, it is possible to complete the colouring by a simple recolouring operation<br />along an \augmenting" path of length O(log n) starting from the<br />uncoloured vertex [17].)<br />It is an open problem whether randomization is necessary in all of the above algorithmic results; the asymptotically best deterministic protocols known to date need O(n(n)) rounds, where (n) tends (very slowly) to<br />zero as the number of vertices grows [1, 18]. In a 1968 paper Vizing asked whether upper bounds for the chromatic<br />number better than those given by Brooks' Theorem existed, provided some sparsity conditions were satised. In particular, he asked what happens for triangle-free graphs. We shall refer to colourings of trianglefree<br />graphs using signicantly fewer than colours as Brooks-Vizing<br />colourings. This existential problem was settled about two decades later. A. Johansson<br />[9] showed that every triangle-free graph has chromatic number<br />O(= log). This is best-possible up to a constant factor, since Bollobas had shown the existence of graphs with arbitrarily high girth such that<br />(G) = (=log) (the girth of a graph G is the size of the smallest<br />cycle therein) [3]. Johansson's result, as well as an earlier result of Kim [14], which shows that graphs of girth at least 5 have chromatic number (1+o(1))= log, make use of certain distributed colouring algorithms, but their results are only existential in the following sense. They show only that the<br />probability that the algorithm produces a valid colouring is positive.<br />Their analyses do not rule out the possibility of their algorithms failing with a high probability. (But then, this was not their main concern.) In this paper, we show that Brooks-Vizing colourings can be computed eciently even in a distributed setting. We present very simple randomized, distributed algorithms, which are also easily implementable on a pram and in the sequential setting, and demonstrate that they<br />produce the desired colourings with high probability.<br />Our algorithms are variants of the dozing-o algorithm. In fact, when the input graph has no 4-cycles (girth 5 or greater) the algorithm is the<br />dozing-o algorithm modied so that the probability that vertices awake is not constant but varies with the round. This probability, initially very low, quickly rises to one, causing the algorithm to be behave as the<br />trivial one from that point on. For the general triangle-free (girth 4) case, the algorithm adds a mechanism which forces the vertex degrees<br />of the uncoloured portion of the graph to remain roughly regular. This regularity is extremely useful in the analysis, but unfortunately gives<br />the algorithm a somewhat high message and space complexity. It may be, however, that this mechanism is unnecessary. The simplicity of the<br />algorithms is an appealing feature and we expect them to work quite well in practice.<br />Although these algorithms display similarities to those in Kim's work [14], we have striven for speed and simplicity. Moreover, our analyses are<br />much simpler than those given there. It is perhaps worth remarking that our analyses demonstrate that Brooks-Vizing colourings are not rare, for the randomized colour assignments computed by the algorithm almost<br />always produces such a colouring. Furthermore, they highlight the role played by the girth assumption. Our result may be conveniently stated as follows: We give an algorithm<br />which, for any triangle-free, D-regular input graph G such that<br />D log1+ n, where &gt; 0 is any xed constant, computes with probability<br />1−o(1) a vertex colouring of G using D=k colours, for any k logD, where is a constant which depends on . Moreover, with probability 1 − o(1), the colouring will be completed within O<br />k + log n logD<br /> rounds in the synchronous, message-passing distributed model of computation<br />(with no shared memory). Both of the above o(1) terms are functions<br />going to 0 with n, the number of vertices in the network.<br />The statement of the theorem allows some flexibility in the choice of k and D. For instance, by choosing D nc= log log n, where c &gt; 1 is any<br />constant, and k = log log n, the algorithm will compute a (D= log log n)-<br />colouring in just O(log log n) rounds. Or, by choosing D nc=<br />p log n the<br />algorithm will compute a (D=<br />p log n)-colouring in O(<br />p log n) rounds. Notice<br />also that the algorithm works for k = (logD), thereby matching the lower bounds of Bollobas and the existential statements of Johansson and Kim. It should be pointed out however that our statement is weaker than<br />their existential statement insofar as it needs the additional assumption D = (logn). This, as well as the regularity assumption (also assumed in [9, 14]), might in fact be an artifact of our analysis which relies on<br />large deviation inequalities that cease to give strong enough bounds for lower values of D. Although we stated our result in its most general form, in this abstract<br />we shall present a slightly weaker version, due to lack of space. Namely, we shall show that the above statement holds with the running timereplaced by O(log n).

For a given number of colours, $s$, the guessing number of a graph is the base $s$ logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the $n$-vertex cycle graph $C_n$ is $n/2$. It is known that the guessing number equals $n/2$ whenever $n$ is even or $s$ is a perfect square. We show that, for any given integer $s \geq 2$, if $a$ is the largest factor of $s$ less than or equal to $\sqrt{s}$, for sufficiently large odd $n$, the guessing number of $C_n$ with $s$ colours is $(n-1)/2 + \log_s(a)$. This answers a question posed by Christofides and Markström in 2011.We also present an explicit protocol which achieves this bound for every $n$. Linking this to index coding with side information, we deduce that the information defect of $C_n$ with $s$ colours is $(n+1)/2 - \log_s(a)$ for sufficiently large odd $n$.

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

The chromatic polynomial $P(G,\lambda)$ gives the number of $\lambda$-colourings of a graph. If $P(G,\lambda)=P(H_{1},\lambda)P(H_{2},\lambda)/P(K_{r},\lambda)$, then the graph $G$ is said to have a chromatic factorisation with chromatic factors $H_{1}$ and $H_{2}$. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic factorisations, but have chromatic polynomials that are not the chromatic polynomial of any clique-separable graph. A certificate of factorisation, that is, a sequence of rewritings based on identities for the chromatic polynomial, is given that explains the chromatic factorisations of graphs from this family. We show that the graphs in this infinite family are the only graphs that have a chromatic factorisation satisfying this certificate and having the odd cycle $C_{2n+1}$, $n\geq2$, as a chromatic factor.

LEGO and Adult Hobbyism For much of its history, LEGO has been regarded as a – if not the – children’s toy. Partially through The LEGO Group (TLG)’s own careful deployment of research on constructivist learning, the building system’s recombinatory logic, bright colours, and foot-destroying durability have become associated with paradigmatic notions of what children’s play is and does (Giddings; Maddalena). And yet the world of adult LEGO hobbyism is complex, rich, and worthy of scholarly attention in its own regard. As recent headlines about the popularity of toys among adults have indicated, LEGO is increasingly viewed as a legitimate adult pastime, if not investment opportunity (Peachey; Fuller and Dorning). Over the course of the pandemic, TLG very carefully targeted whole product lines towards adult builders. Themes like Architecture, Creator Expert, Art, and Botanicals are marketed as tools for adult mindfulness and meditation: expensive 3D colouring books that, when finished, belong on walls, office desks, or coffee tables as display pieces, rather than ending up in pieces on the floor. Such sets may even be accompanied by Spotify playlists meant to serve as ambient noise during building (for more on this topic, see Ogden). But LEGO has been used, collected, and modified by adult enthusiasts for decades, spawning the large and vibrant community of self-identified Adult Fans of LEGO, or AFOLs (Jennings). LEGO serves as the medium for a wide and surprisingly varied matrix of practices undertaken quite seriously by these committed hobbyists, as well as scores of artists and entrepreneurs – including those whose building and collecting practices put them at odds with the AFOL community (Taylor and Ingraham). These practices include, but are not limited to, crafting and circulating instructions for fan-made creations (My Own Creation, or MOCs), either as a hobby or a lucrative business; creating LEGO sculptures for private collectors, museums, fan conventions, and art galleries; designing and printing LEGO-compatible pieces and minifigure accessories that LEGO itself cannot or will not create; and modifying minifigures with decals and/or commercial-quality printing, most often to resemble characters from media franchises for which LEGO does not have licensing agreements. Given how expensive LEGO is, and how difficult it can be to acquire individual pieces through LEGO’s own Website (particularly in bulk), almost all these practices rely on ready access to aftermarket sources of LEGO. While Facebook and eBay are common resources for used or pre-purchased sets, Bricklink is the most popular site for serious builders. Dubbed the ‘LEGO eBay’, it provides a platform for hundreds of thousands of individual sellers from all over the world to make their pieces available to millions of buyers. In Bricklink’s cataloguing system, every type of piece is given a part number (the same as LEGO uses), and the site provides the piece breakdown for (almost) every set; every Bricklink seller is required to catalogue their own inventory using this system. As a result, buyers can search for and purchase any piece that LEGO has made, in any colour available, either used or new, and at quantity. Because these are sourced from individual sellers and not LEGO directly, and both buyers and sellers can rate each other’s service (as on any good platform), Bricklink works far better than LEGO (or Facebook and eBay, for that matter) to service the kinds of bulk and specialised piece purchases required by both dabblers and serious hobbyists. If you’ve lost pieces required to assemble a set, look for them on Bricklink, where you can sort by brick condition, country of seller, and number of available pieces; if you want to collect Star Wars-themed minifigures, but don’t care for the grey-monotone spaceships they come with, itemise and inventory the unwanted pieces and sell them on your Bricklink storefront. You can also browse for whole sets, in new or used condition; official instructions; and instructions for user-created MOCs. You can even purchase the colourful cardboard boxes (emptied) in which LEGO sets come packaged. In the conversations my colleagues and I had with LEGO hobbyists, artists, and entrepreneurs, which formed the basis for our edited volume on LEGO as material medium, it became clear that Bricklink features centrally in their creative practices (Taylor and Ingraham). It constitutes a vital infrastructure for the teeming exchange in aftermarket LEGO, amounting to millions of transactions and billions of plastic pieces a year. It is, as one influential LEGO blogger put it, the “lifeblood” of the hobbyist community (Ong). Acknowledging Bricklink’s vital role among adult LEGO enthusiasts, this article takes up the implications of its acquisition by TLG in 2019, and the company’s subsequent efforts to curtail what it regards as unsanctioned uses of its product. Where journalistic coverage largely focussed on the ambivalence with which the AFOL community regards Bricklink’s acquisition (Wood), I turn instead to the effects that it has had (and may yet have) on the artists and entrepreneurs for whom LEGO is not just a creative medium, but a livelihood. To do so, I approach LEGO as a media platform, one engaged – as other commercial platforms are – in extending monopolistic reach over the means through which we craft and exchange cultural productions. LEGO as Platform Since losing its legal monopoly over interlocking bricks, allowing for hobbyists and other toy manufacturers to create products that are LEGO-compatible, TLG has pursued other means to dictate how its products are used (Rimmer). Namely, it has grown and leveraged its power as a media platform, using a combination of technical, discursive, and infrastructural techniques to shape how users produce and circulate their LEGO creations. In understanding LEGO as a media platform, I am more concerned with the ways in which the toy construction system operates as an apparatus of creation and connection, rather than its transmedial reach across movies, video games, television shows, and so on (which is how LEGO is more frequently discussed by media scholars; see Hains and Mazzarella). Tarleton Gillespie’s generative theorisation of platforms is useful in this regard, for the ways in which it deconstructs the semantic richness of the term and shows the ways in which media companies traffic in these manifold meanings. Gillespie speaks of the “figurative” sense of platforms (as structures that enable meaningful activity); the “computational” sense (as software operating systems); the “architectural” sense (the oldest and most mundane use of the term, as a technique of physical elevation); and in the political sense, as an organisation’s core values (Gillespie 349–50). As a “materially digital” building system, LEGO operationalises all these senses (Maddalena). It is at once a physical computational platform, particularly if we consider the numerous product lines as different applications; a tool for supporting creative expression, both in tangible and discursive ways (architectural and figurative platform, respectively); and a company that very publicly engages in progressivist education and safely progressive politics. For the LEGO hobbyists and entrepreneurs mentioned above, those who are most invested in LEGO as a medium of expression rather than as an education tool, the architectural and computational aspects of LEGO are most immediate. LEGO is their medium of choice, the recombinatory potentials of its elements making it possible to translate virtually any experience or artifact into LEGO form: in other words, to “LEGOfy” it (Ingraham and Taylor). These creators have come up with numerous applications of the toy, some mentioned above (and documented more exhaustively in our other work), from minifigure modification to self-published MOCs. Such applications frequently undermine the careful work carried out by TLG to position LEGO as a progressive and family-oriented educational tool: as a figurative and political platform for constructive play. Below are two examples of small businesses whose products created for adult LEGO enthusiasts produce tensions between these various facets of LEGO’s platform. Making Guns and Breaking Bad BrickArms manufactures small arms to fit in the small arms of LEGO’s iconic minifigures: scale models of actual historical or contemporary weapons, as well as some science fiction-themed killing machines. In glib terms, BrickArms makes guns, many of them, for LEGO minifigures. Their products are used extensively by hobbyists and other entrepreneurs because they fill a void left by LEGO’s consistently stated (if inconsistently exercised) stance against military weaponry. BrickArms does not sell its products directly; rather, they are sold through other small businesses like the military-themed set maker Brickmania, which also features BrickArms weapons in its products. If BrickArms undermines LEGO’s figurative platform by producing a range of realistic weapons that are at once technically interoperable and ideologically incompatible with the toy, custom minifigure makers like Citizen Brick (which also sells BrickArms products on its Website) pose a related, but different threat. Citizen Brick produces adult-themed minifigures, accessories, and pieces by imprinting official LEGO elements with designs that are frequently raunchy, violent, and/or flagrantly in violation both of TLG’s intellectual property arrangements with other media franchises, as well as its family-friendly brand image. For instance, one popular Citizen Brick product is the “Chemistry Enthusiast”, a minifigure which bears a striking (albeit LEGOfied) resemblance to science teacher-turned-meth-magnate Walter White from Breaking Bad. Both Citizen Brick and BrickArms operate legally, producing LEGO-compatible products, but both quite deliberately (even gleefully, in the case of Citizen Brick and its tongue-in-cheek marketing) undermine TLG’s core values. These and other similar businesses, not to mention countless hobbyist MOC-makers, present TLG with a conundrum: how to stop entrepreneurs, hobbyists, and artists from creating and distributing violent, reactionary, and/or non-family-friendly uses of a product that is otherwise celebrated for its limitless expressive potential? To put it in terms familiar to media theorists: how might TLG, as platform owner, moderate undesirable content – in this case, content that is tactile and material rather than virtual? Re-Assembling Bricklink Content moderation may not have been the sole reason between TLG’s decision to purchase Bricklink from South Korean tech business Nexon; certainly, gaining access to the data on users’ inventories and transactions was also among its key considerations. But the actions taken by TLG shortly after it took control suggest that curbing undesirable uses of its product (particularly those carried out for commercial purposes) was at least one major goal. While TLG rationalised its purchase of Bricklink in terms of supporting its users (“empower the creativity of AFOLs and fuel future innovations”), its first steps were to ban products that featured builds of characters and scenes from media franchises outside of LEGO’s current license partnerships (Bricklink). It also banned the circulation of user-generated LEGO-interoperable products; notable among these banned products were those made by BrickArms. TLG’s rationale for banning BrickArms was that LEGOfied assault rifles are incommensurate with the values of LEGO – that is, with its status as a figurative and political platform. This is consistent with its public statements (published in 2010, and no longer available online) regarding its refusal to produce certain tools of violence for its own product lines: “the basic aim is to avoid realistic weapons and military equipment that children may recognize from hot spots around the world and to refrain from showing violent or frightening situations when communicating about LEGO products” (TLG, quoted by Lendon). As laudable as this corporate stance is, TLG’s weapons ban has been inconsistently exercised; after all, “realistic” weapons from swords to submachine guns appear regularly across LEGO’s multiple product lines built around fantasy-themed violence. A different, perhaps more compelling rationale for TLG’s acquisition of Bricklink, and its ban both on non-LEGO products on the site (not just guns) and on products modelled after unlicenced media franchises, is to accomplish through infrastructural means what the company has been unable to do legally, since losing its suit against Canadian toymaker Mega Bloks: that is, to assert a monopoly over building toy systems by curtailing the capacity for either businesses or private individuals to incorporate non-LEGO products into their creations, whether commercial and non-commercial. Like other major media platforms, LEGO encourages connection, openness, and creativity – so long as we use its platform, and its platform exclusively, to do so. Here, we can mobilise a further notion of platform, which has attracted considerable scholarly attention in the last decade: understanding platforms as the economic engine of contemporary late-stage capitalism, with the “platformisation” of increasing sectors of the economy entailing the transformation of jobs and, perhaps as frequently, hobbies, into gig work (Nieborg and Poell; Vallas and Schor). Under these conditions, the platformisation of LEGO – facilitated, in part, by TLG’s acquisition of Bricklink – positions small businesses like BrickArms and Citizen Brick, not to mention the countless artists who work with LEGO, as platform workers. They are utterly dependent on access to LEGO, both in terms of the interoperability of their products and in terms of their ability to distribute their products to LEGO hobbyists and enthusiasts. Like other platform workers, their livelihoods are profoundly shaped by the regulatory regimes and policy shifts of corporate media giants. Conclusion: Infrastructural Instructions While subtle and arguably relatively contained in its effects, TLG’s operations regarding its acquisition and subsequent content moderation of Bricklink align it with other “platformised infrastructures” like Facebook, Twitter, and Steam: commercial-run systems of connection that purport to foster open, creative forms of production and exchange, while at the same time extending near-monopolistic control over those means of production and exchange (Plantin et al. 298). TLG wants us all to play nice, and very adroitly positions LEGO as the paradigmatic medium for children’s open exploratory play and, increasingly, for adults’ mindfulness and self-care. It has very clear ideas about what playing nice entails: no offensive content, nothing overtly harmful or hurtful, and a cheerful embrace of a sort of focus-grouped politics of progressive representation (Johnson). But playing nice also means, crucially, avoiding anything that is not LEGO. It is a notion of nice that is fundamentally commensurate with and concerned for LEGO’s virtual monopoly on recombinatory, material play. This is a monopoly which, while no longer legal, has been waged ideologically to great success, such that incorporating LEGO-compatible, non-LEGO-branded building bricks into one’s hobbyist builds or commercial aftermarket products is anathema to the AFOL community – those for whom Bricklink is their lifeblood (Taylor). With the acquisition of Bricklink, it is now a monopoly that can be exerted infrastructurally as well. For those who rely on Bricklink for their hobby, if not their livelihood, the message is clear: play by our rules, or don’t play at all. LEGO famously has a profoundly ambivalent relationship to instructions: instructions formalise and cement LEGO’s creative potentials, but also curtail them. By way of conclusion, we might consider how LEGO’s acquisition of Bricklink constitutes a set of infrastructural instructions: prescriptions not for how certain pieces can fit together to build certain things, but around what constitutes appropriate and acceptable uses of a product that ostensibly has limitless creative possibilities. This set of infrastructural prescriptions has less to do with LEGO’s moral stance as an arbiter and champion of creativity, problem-solving, and progressive education – that is, with its operations as a figurative and political platform – and more with LEGO’s monopolistic aspirations to be ‘the’ operating platform for materially digital creation. References BrickLink. “BrickLink Joins the LEGO Group – FAQ.” 26 Jan. 2023 <https://www.bricklink.com/r3/announcement/lego_bl_faq.page>. Fuller, Jason, and Courtney Dorning. “It May Be More Lucrative to Invest in Collectible LEGO Sets than in Gold, Study Finds.” NPR, 21 Dec. 2021. <https://www.npr.org/2021/12/21/1066493441/it-may-be-more-lucrative-to-invest-in-collectible-lego-sets-than-in-gold-study-f>. Giddings, Seth. “Bright Bricks, Dark Play: On the Impossibility of Studying LEGO.” In LEGO Studies: Examining the Building Blocks of a Transmedial Phenomenon. Ed. Mark J.P. Wolf. New York: Routledge, 2014. 241–67. <https://eprints.soton.ac.uk/386083/>. Gillespie, Tarleton. “The Politics of ‘Platforms.’” New Media & Society 12.3 (2010): 347–64. <https://doi.org/10.1177/1461444809342738>. Hains, Rebecca C., and Sharon R. Mazzarella, eds. Cultural Studies of LEGO: More than Just Bricks. Springer Nature, 2019. Ingraham, Chris, and Nicholas T. Taylor. “Theorybuilding with LEGO: A Material Digital Media.” Digital Doxa, 30 Mar. 2020. <https://www.digitaldoxa.org/post/theorybuilding-with-lego-a-material-digital-media-chris-ingraham-and-nick-taylor>. Jennings, Nancy A. “‘It’s All about the Brick’: Mobilizing Adult Fans of LEGO.” In Cultural Studies of LEGO: More than Just Bricks. Eds. Rebecca C. Hains and Sharon R. Mazzarella. Springer, 2019. 221–43. <https://doi.org/10.1007/978-3-030-32664-7_10>. Johnson, Derek. “A License to Diversify: Media Franchising and the Transformation of the ‘Universal’ LEGO Minifigure.” In Cultural Studies of LEGO: More than Just Bricks. Eds. Rebecca C. Hains and Sharon R. Mazzarella. Springer, 2019. Lendon, Brad. “LEGO Won’t Make Modern War Machines, But Others Are Picking Up the Pieces.” CNN. 26 Jan. 2023 <https://www.cnn.com/style/article/lego-military-sets-intl-hnk-dst/index.html>. Maddalena, Kate. Mediating Atomistic Ontologies: LEGO, Synthetic Biology, and a Digital Episteme. North Carolina State University, 15 Apr. 2014. <http://www.lib.ncsu.edu/resolver/1840.16/9473>. Nieborg, David B., and Thomas Poell. “The Platformization of Cultural Production: Theorizing the Contingent Cultural Commodity.” New Media & Society 20.11 (2018): 4275–92. <https://doi.org/10.1177/1461444818769694>. Ogden, Malcolm. “LEGOfied Sound: On the Labor and Leisure of ‘LEGO White Noise.’” Resonance: The Journal of Sound and Culture 4.2 (2023). Ong, Jay. “LEGO Acquires Bricklink. Why I Think This Is a Bad Idea.” Jay’s Brick Blog, 26 Nov. 2019. <https://jaysbrickblog.com/news/lego-acquires-bricklink-why-i-think-this-is-a-bad-idea/>. Peachey, Kevin. “Parents Buying More Toys and Games for Themselves.” BBC News, 25 Jan. 2023. <https://www.bbc.com/news/business-64386885>. Plantin, Jean-Christophe, et al. “Infrastructure Studies Meet Platform Studies in the Age of Google and Facebook.” New Media & Society 20.1 (2018): 293–310. <https://doi.org/10.1177/1461444816661553>. Rimmer, Matthew. “Trouble in Legoland: Trade Mark Law and Functionality.” Australian Intellectual Property Newsletter Archive, 22 Mar. 2006. <https://eprints.qut.edu.au/215082/>. Taylor, Nicholas T. “Purity and the Boundaries of Belonging.” In LEGOfied: Building Blocks as Media. Eds. Nicholas T. Taylor and Chris Ingraham. Bloomsbury Academic, 2020. 139–64. <https://doi.org/10.5040/9781501354076>. Taylor, Nicholas T., and Chris Ingraham, eds. LEGOfied: Building Blocks as Media. Bloomsbury Publishing, 2020. Vallas, Steven, and Juliet B. Schor. “What Do Platforms Do? Understanding the Gig Economy.” Annual Review of Sociology 46.1 (2020): 273–94. <https://doi.org/10.1146/annurev-soc-121919-054857>. Wood, Zoe. “Lego Accused of Muscling In on Fans after BrickLink Takeover.” The Guardian, 20 Dec. 2019. <https://www.theguardian.com/lifeandstyle/2019/dec/20/lego-accused-of-muscling-in-on-fans-after-bricklink-takeover>.

Over the past century, many books for general readers have styled sharks as “monsters of the deep” (Steele). In recent decades, however, at least some writers have also turned to representing how sharks are seriously threatened by human activities. At a time when media coverage of shark sightings seems ever increasing in Australia, scholarship has begun to consider people’s attitudes to sharks and how these are formed, investigating the representation of sharks (Peschak; Ostrovski et al.) in films (Le Busque and Litchfield; Neff; Schwanebeck), newspaper reports (Muter et al.), and social media (Le Busque et al., “An Analysis”). My own research into representations of surfing and sharks in Australian writing (Brien) has, however, revealed that, although reporting of shark sightings and human-shark interactions are prominent in the news, and sharks function as vivid and commanding images and metaphors in art and writing (Ellis; Westbrook et al.), little scholarship has investigated their representation in Australian books published for a general readership. While recognising representations of sharks in other book-length narrative forms in Australia, including Australian fiction, poetry, and film (Ryan and Ellison), this enquiry is focussed on non-fiction books for general readers, to provide an initial review. Sampling holdings of non-fiction books in the National Library of Australia, crosschecked with Google Books, in early 2021, this investigation identified 50 Australian books for general readers that are principally about sharks, or that feature attitudes to them, published from 1911 to 2021. Although not seeking to capture all Australian non-fiction books for general readers that feature sharks, the sampling attempted to locate a wide range of representations and genres across the time frame from the earliest identified text until the time of the survey. The books located include works of natural and popular history, travel writing, memoir, biography, humour, and other long-form non-fiction for adult and younger readers, including hybrid works. A thematic analysis (Guest et al.) of the representation of sharks in these texts identified five themes that moved from understanding sharks as fishes to seeing them as monsters, then prey, and finally to endangered species needing conservation. Many books contained more than one theme, and not all examples identified have been quoted in the discussion of the themes below. Sharks as Part of the Natural Environment Drawing on oral histories passed through generations, two memoirs (Bradley et al.; Fossa) narrate Indigenous stories in which sharks play a central role. These reveal that sharks are part of both the world and a wider cosmology for Aboriginal and Torres Strait Islander people (Clua and Guiart). In these representations, sharks are integrated with, and integral to, Indigenous life, with one writer suggesting they are “creator beings, ancestors, totems. Their lifecycles reflect the seasons, the landscape and sea country. They are seen in the movement of the stars” (Allam). A series of natural history narratives focus on zoological studies of Australian sharks, describing shark species and their anatomy and physiology, as well as discussing shark genetics, behaviour, habitats, and distribution. A foundational and relatively early Australian example is Gilbert P. Whitley’s The Fishes of Australia: The Sharks, Rays, Devil-fish, and Other Primitive Fishes of Australia and New Zealand, published in 1940. Ichthyologist at the Australian Museum in Sydney from the early 1920s to 1964, Whitley authored several books which furthered scientific thought on sharks. Four editions of his Australian Sharks were published between 1983 and 1991 in English, and the book is still held in many libraries and other collections worldwide. In this text, Whitley described a wide variety of sharks, noting shared as well as individual features. Beautiful drawings contribute information on shape, colouring, markings, and other recognisable features to assist with correct identification. Although a scientist and a Fellow and then President of the Royal Zoological Society of New South Wales, Whitley recognised it was important to communicate with general readers and his books are accessible, the prose crisp and clear. Books published after this text (Aiken; Ayling; Last and Stevens; Tricas and Carwardine) share Whitley’s regard for the diversity of sharks as well as his desire to educate a general readership. By 2002, the CSIRO’s Field Guide to Australian Sharks & Rays (Daley et al.) also featured numerous striking photographs of these creatures. Titles such as Australia’s Amazing Sharks (Australian Geographic) emphasise sharks’ unique qualities, including their agility and speed in the water, sensitive sight and smell, and ability to detect changes in water pressure around them, heal rapidly, and replace their teeth. These books also emphasise the central role that sharks play in the marine ecosystem. There are also such field guides to sharks in specific parts of Australia (Allen). This attention to disseminating accurate zoological information about sharks is also evident in books written for younger readers including very young children (Berkes; Kear; Parker and Parker). In these and other similar books, sharks are imaged as a central and vital component of the ocean environment, and the narratives focus on their features and qualities as wondrous rather than monstrous. Sharks as Predatory Monsters A number of books for general readers do, however, image sharks as monsters. In 1911, in his travel narrative Peeps at Many Lands: Australia, Frank Fox describes sharks as “the most dangerous foes of man in Australia” (23) and many books have reinforced this view over the following century. This can be seen in titles that refer to sharks as dangerous predatory killers (Fox and Ruhen; Goadby; Reid; Riley; Sharpe; Taylor and Taylor). The covers of a large proportion of such books feature sharks emerging from the water, jaws wide open in explicit homage to the imaging of the monster shark in the film Jaws (Spielberg). Shark!: Killer Tales from the Dangerous Depths (Reid) is characteristic of books that portray encounters with sharks as terrifying and dramatic, using emotive language and stories that describe sharks as “the world’s most feared sea creature” (47) because they are such “highly efficient killing machines” (iv, see also 127, 129). This representation of sharks is also common in several books for younger readers (Moriarty; Rohr). Although the risk of being injured by an unprovoked shark is extremely low (Chapman; Fletcher et al.), fear of sharks is prevalent and real (Le Busque et al., “People’s Fear”) and described in a number of these texts. Several of the memoirs located describe surfers’ fear of sharks (Muirhead; Orgias), as do those of swimmers, divers, and other frequent users of the sea (Denness; de Gelder; McAloon), even if the author has never encountered a shark in the wild. In these texts, this fear of sharks is often traced to viewing Jaws, and especially to how the film’s huge, bloodthirsty great white shark persistently and determinedly attacks its human hunters. Pioneer Australian shark expert Valerie Taylor describes such great white sharks as “very big, powerful … and amazingly beautiful” but accurately notes that “revenge is not part of their thought process” (Kindle version). Two books explicitly seek to map and explain Australians’ fear of sharks. In Sharks: A History of Fear in Australia, Callum Denness charts this fear across time, beginning with his own “shark story”: a panicked, terror-filled evacuation from the sea, following the sighting of a shadow which turned out not to be a shark. Blake Chapman’s Shark Attacks: Myths, Misunderstandings and Human Fears explains commonly held fearful perceptions of sharks. Acknowledging that sharks are a “highly emotive topic”, the author of this text does not deny “the terror [that] they invoke in our psyche” but makes a case that this is “only a minor characteristic of what makes them such intriguing animals” (ix). In Death by Coconut: 50 Things More Dangerous than a Shark and Why You Shouldn’t Be Afraid of the Ocean, Ruby Ashby Orr utilises humour to educate younger readers about the real risk humans face from sharks and, as per the book’s title, why they should not be feared, listing champagne corks and falling coconuts among the many everyday activities more likely to lead to injury and death in Australia than encountering a shark. Taylor goes further in her memoir – not only describing her wonder at swimming with these creatures, but also her calm acceptance of the possibility of being injured by a shark: "if we are to be bitten, then we are to be bitten … . One must choose a life of adventure, and of mystery and discovery, but with that choice, one must also choose the attendant risks" (2019: Kindle version). Such an attitude is very rare in the books located, with even some of the most positive about these sea creatures still quite sensibly fearful of potentially dangerous encounters with them. Sharks as Prey There is a long history of sharks being fished in Australia (Clark). The killing of sharks for sport is detailed in An American Angler in Australia, which describes popular adventure writer Zane Grey’s visit to Australia and New Zealand in the 1930s to fish ‘big game’. This text includes many bloody accounts of killing sharks, which are justified with explanations about how sharks are dangerous. It is also illustrated with gruesome pictures of dead sharks. Australian fisher Alf Dean’s biography describes him as the “World’s Greatest Shark Hunter” (Thiele), this text similarly illustrated with photographs of some of the gigantic sharks he caught and killed in the second half of the twentieth century. Apart from being killed during pleasure and sport fishing, sharks are also hunted by spearfishers. Valerie Taylor and her late husband, Ron Taylor, are well known in Australia and internationally as shark experts, but they began their careers as spearfishers and shark hunters (Taylor, Ron Taylor’s), with the documentary Shark Hunters gruesomely detailing their killing of many sharks. The couple have produced several books that recount their close encounters with sharks (Taylor; Taylor, Taylor and Goadby; Taylor and Taylor), charting their movement from killers to conservationists as they learned more about the ocean and its inhabitants. Now a passionate campaigner against the past butchery she participated in, Taylor’s memoir describes her shift to a more respectful relationship with sharks, driven by her desire to understand and protect them. In Australia, the culling of sharks is supposedly carried out to ensure human safety in the ocean, although this practice has long been questioned. In 1983, for instance, Whitley noted the “indiscriminate” killing of grey nurse sharks, despite this species largely being very docile and of little threat to people (Australian Sharks, 10). This is repeated by Tony Ayling twenty-five years later who adds the information that the generally harmless grey nurse sharks have been killed to the point of extinction, as it was wrongly believed they preyed on surfers and swimmers. Shark researcher and conservationist Riley Elliott, author of Shark Man: One Kiwi Man’s Mission to Save Our Most Feared and Misunderstood Predator (2014), includes an extremely critical chapter on Western Australian shark ‘management’ through culling, summing up the problems associated with this approach: it seems to me that this cull involved no science or logic, just waste and politics. It’s sickening that the people behind this cull were the Fisheries department, which prior to this was the very department responsible for setting up the world’s best acoustic tagging system for sharks. (Kindle version, Chapter 7) Describing sharks as “misunderstood creatures”, Orr is also clear in her opposition to killing sharks to ‘protect’ swimmers noting that “each year only around 10 people are killed in shark attacks worldwide, while around 73 million sharks are killed by humans”. She adds the question and answer, “sounds unfair? Of course it is, but when an attack is all over the news and the people are baying for shark blood, it’s easy to lose perspective. But culling them? Seriously?” (back cover). The condemnation of culling is also evident in David Brooks’s recent essay on the topic in his collection of essays about animal welfare, conservation and the relationship between humans and other species, Animal Dreams. This disapproval is also evident in narratives by those who have been injured by sharks. Navy diver Paul de Gelder and surfer Glen Orgias were both bitten by sharks in Sydney in 2009 and both their memoirs detail their fear of sharks and the pain they suffered from these interactions and their lengthy recoveries. However, despite their undoubted suffering – both men lost limbs due to these encounters – they also attest to their ongoing respect for these creatures and specify a shared desire not to see them culled. Orgias, instead, charts the life story of the shark who bit him alongside his own story in his memoir, musing at the end of the book, not about himself or his injury, but about the fate of the shark he had encountered: great whites are portrayed … as pathological creatures, and as malevolent. That’s rubbish … they are graceful, mighty beasts. I respect them, and fear them … [but] the thought of them fighting, dying, in a net upsets me. I hope this great white shark doesn’t end up like that. (271–271) Several of the more recent books identified in this study acknowledge that, despite growing understanding of sharks, the popular press and many policy makers continue to advocate for shark culls, these calls especially vocal after a shark-related human death or injury (Peppin-Neff). The damage to shark species involved caused by their killing – either directly by fishing, spearing, finning, or otherwise hunting them, or inadvertently as they become caught in nets or affected by human pollution of the ocean – is discussed in many of the more recent books identified in this study. Sharks as Endangered Alongside fishing, finning, and hunting, human actions and their effects such as beach netting, pollution and habitat change are killing many sharks, to the point where many shark species are threatened. Several recent books follow Orr in noting that an estimated 100 million sharks are now killed annually across the globe and that this, as well as changes to their habitats, are driving many shark species to the status of vulnerable, threatened or towards extinction (Dulvy et al.). This is detailed in texts about biodiversity and climate change in Australia (Steffen et al.) as well as in many of the zoologically focussed books discussed above under the theme of “Sharks as part of the natural environment”. The CSIRO’s Field Guide to Australian Sharks & Rays (Daley et al.), for example, emphasises not only that several shark species are under threat (and protected) (8–9) but also that sharks are, as individuals, themselves very fragile creatures. Their skeletons are made from flexible, soft cartilage rather than bone, meaning that although they are “often thought of as being incredibly tough; in reality, they need to be handled carefully to maximise their chance of survival following capture” (9). Material on this theme is included in books for younger readers on Australia’s endangered animals (Bourke; Roc and Hawke). Shark Conservation By 1991, shark conservation in Australia and overseas was a topic of serious discussion in Sydney, with an international workshop on the subject held at Taronga Zoo and the proceedings published (Pepperell et al.). Since then, the movement to protect sharks has grown, with marine scientists, high-profile figures and other writers promoting shark conservation, especially through attempts to educate the general public about sharks. De Gelder’s memoir, for instance, describes how he now champions sharks, promoting shark conservation in his work as a public speaker. Peter Benchley, who (with Carl Gottlieb) recast his novel Jaws for the film’s screenplay, later attested to regretting his portrayal of sharks as aggressive and became a prominent spokesperson for shark conservation. In explaining his change of heart, he stated that when he wrote the novel, he was reflecting the general belief that sharks would both seek out human prey and attack boats, but he later discovered this to be untrue (Benchley, “Without Malice”). Many recent books about sharks for younger readers convey a conservation message, underscoring how, instead of fearing or killing sharks, or doing nothing, humans need to actively assist these vulnerable creatures to survive. In the children’s book series featuring Bindi Irwin and her “wildlife adventures”, there is a volume where Bindi and a friend are on a diving holiday when they find a dead shark whose fin has been removed. The book not only describes how shark finning is illegal, but also how Bindi and friend are “determined to bring the culprits to justice” (Browne). This narrative, like the other books in this series, has a dual focus; highlighting the beauty of wildlife and its value, but also how the creatures described need protection and assistance. Concluding Discussion This study was prompted by the understanding that the Earth is currently in the epoch known as the Anthropocene, a time in which humans have significantly altered, and continue to alter, the Earth by our activities (Myers), resulting in numerous species becoming threatened, endangered, or extinct. It acknowledges the pressing need for not only natural science research on these actions and their effects, but also for such scientists to publish their findings in more accessible ways (see, Paulin and Green). It specifically responds to demands for scholarship outside the relevant areas of science and conservation to encourage widespread thinking and action (Mascia et al.; Bennett et al.). As understanding public perceptions and overcoming widely held fear of sharks can facilitate their conservation (Panoch and Pearson), the way sharks are imaged is integral to their survival. The five themes identified in this study reveal vastly different ways of viewing and writing about sharks. These range from seeing sharks as nothing more than large fishes to be killed for pleasure, to viewing them as terrifying monsters, to finally understanding that they are amazing creatures who play an important role in the world’s environment and are in urgent need of conservation. This range of representation is important, for if sharks are understood as demon monsters which hunt humans, then it is much more ‘reasonable’ to not care about their future than if they are understood to be fascinating and fragile creatures suffering from their interactions with humans and our effect on the environment. Further research could conduct a textual analysis of these books. In this context, it is interesting to note that, although in 1949 C. Bede Maxwell suggested describing human deaths and injuries from sharks as “accidents” (182) and in 2013 Christopher Neff and Robert Hueter proposed using “sightings, encounters, bites, and the rare cases of fatal bites” (70) to accurately represent “the true risk posed by sharks” to humans (70), the majority of the books in this study, like mass media reports, continue to use the ubiquitous and more dramatic terminology of “shark attack”. The books identified in this analysis could also be compared with international texts to reveal and investigate global similarities and differences. While the focus of this discussion has been on non-fiction texts, a companion analysis of representation of sharks in Australian fiction, poetry, films, and other narratives could also be undertaken, in the hope that such investigations contribute to more nuanced understandings of these majestic sea creatures. 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