Relevant bibliographies by topics / Odd colouring

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Academic literature on the topic 'Odd colouring'

Author: Grafiati
Published: 25 May 2024

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Journal articles on the topic "Odd colouring"

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KANG, ROSS J., and FRANÇOIS PIROT. "Distance Colouring Without One Cycle Length." Combinatorics, Probability and Computing 27, no. 5 (March 25, 2018): 794–807. http://dx.doi.org/10.1017/s0963548318000068.

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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).
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ZELENYUK, YEVHEN, and YULIYA ZELENYUK. "COUNTING SYMMETRIC BRACELETS." Bulletin of the Australian Mathematical Society 89, no. 3 (August 22, 2013): 431–36. http://dx.doi.org/10.1017/s0004972713000701.

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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.
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Kang, Dong Yeap, and Sang-Il Oum. "Improper colouring of graphs with no odd clique minor." Combinatorics, Probability and Computing 28, no. 5 (February 4, 2019): 740–54. http://dx.doi.org/10.1017/s0963548318000548.

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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.
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KAWARABAYASHI, KEN-ICHI. "A Weakening of the Odd Hadwiger's Conjecture." Combinatorics, Probability and Computing 17, no. 6 (November 2008): 815–21. http://dx.doi.org/10.1017/s0963548308009462.

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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].
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BOUSQUET, NICOLAS, LOUIS ESPERET, ARARAT HARUTYUNYAN, and RÉMI DE JOANNIS DE VERCLOS. "Exact Distance Colouring in Trees." Combinatorics, Probability and Computing 28, no. 2 (July 24, 2018): 177–86. http://dx.doi.org/10.1017/s0963548318000378.

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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.
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Bryant, Darryn, and C. A. Rodger. "On the completion of latin rectangles to symmetric latin squares." Journal of the Australian Mathematical Society 76, no. 1 (February 2004): 109–24. http://dx.doi.org/10.1017/s1446788700008739.

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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.
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KINNERSLEY, WILLIAM B., KEVIN G. MILANS, and DOUGLAS B. WEST. "Degree Ramsey Numbers of Graphs." Combinatorics, Probability and Computing 21, no. 1-2 (February 2, 2012): 229–53. http://dx.doi.org/10.1017/s0963548311000617.

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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.
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Stone, Phil. "Non-Mathematical Musings on Information Theory and Networked Musical Practice." Organised Sound 26, no. 3 (December 2021): 327–32. http://dx.doi.org/10.1017/s1355771821000418.

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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.
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Jayawardene, C. J., J. N. Senadheera, K. A. S. N. Fernando, and W. C. W. Navaratna. "On Star-critical (K1,n, K1,m + e) Ramsey Numbers." Annals of Pure and Applied Mathematics 22, no. 02 (2020): 75–82. http://dx.doi.org/10.22457/apam.v22n2a02702.

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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.
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Chistikov, Dmitry, Olga Goulko, Adrian Kent, and Mike Paterson. "Globe-hopping." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2238 (June 2020): 20200038. http://dx.doi.org/10.1098/rspa.2020.0038.

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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.
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Dissertations / Theses on the topic "Odd colouring"

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Ouyang, Qiancheng. "Some colouring problems in edge/vertex-coloured graphs : Structural and extremal studies." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPASG060.

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La coloration de graphes est l'un des sujets les plus connus, populaires et largement étudiés dans le domaine de la théorie des graphes, avec une vaste littérature comprenant des approches provenant de nombreux domaines ainsi que de nombreux problèmes qui sont encore ouverts et étudiés par divers mathématiciens et informaticiens à travers le monde. Le Problème des Quatre Couleurs, à l'origine de l'étude de la coloration des graphes, a été l'un des problèmes centraux en théorie des graphes au siècle dernier. Il demande s'il est possible de colorer proprement chaque graphe planaire avec quatre couleurs. Malgré son origine théorique, la coloration des graphes a trouvé de nombreuses applications pratiques telles que la planification, les problèmes d'assignation de fréquences, la segmentation, etc. Le Problème des Quatre Couleurs est l'un des problèmes importants parmi de nombreux problèmes de la théorie des graphes chromatiques, à partir duquel de nombreuses variantes et généralisations ont été proposées. Tout d'abord, dans cette thèse, nous visons à optimiser la stratégie de coloration des sommets de graphes et d'hypergraphes avec certaines contraintes données, en combinant le concept de coloration propre et d'élément représentatif de certains sous-ensembles de sommets. D'autre part, en fonction du sujet à colorer, une grande quantité de recherches et de problèmes de graphes à arêtes colorées ont émergé, avec des applications importantes en biologie et en technologies web. Nous fournissons quelques résultats analogues pour certaines questions de connectivité, afin de décrire des graphes dont les arêtes sont attribuées suffisamment de couleurs, garantissant ainsi des arbres couvrants ou des cycles ayant une structure chromatique spécifique
Graph colouring is one of the best known, popular and extensively researched subject in the field of graph theory, having a wide literature with approaches from many domains and a lot of problems, which are still open and studied by various mathematicians and computer scientists along the world. The Four Colour Problem, originating the study of graph colouring, was one of the central problem in graph theory in the last century, which asks if it is possible to colour every planar graph properly by four colours. Despite the theoretical origin, the graph colouring has found many applications in practice like scheduling, frequency assignment problems, segmentation, etc. The Four Colour Problem is a significant one among many problems in chromatic graph theory, from which many variants and generalizations have been proposed. Firstly, in this thesis, we aim to optimize the strategy to colour the vertex of graphs and hypergraphs with some given constraints, which combines the concept of proper colouring and representative element of some vertex subsets. On the other hand, according to the subject to be coloured, a large amount of research and problems of edge-coloured graphs have emerged, which have important applications to biology and web technologies. We provide some analogous results for some connectivity issues—to describe graphs whose edges are assigned enough colours, that guarantee spanning trees or cycles of a specific chromatic structure
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Books on the topic "Odd colouring"

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Barnes, Jaxson. Odd Squad Waves Diagonals Lines Swirls Dots Coloring Book: Awesome Illustrations Spirograph Styles Colouring Books for Adult. Independently Published, 2021.

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Barnes, Jaxson. Odd Squad Waves Diagonals Lines Swirls Dots Coloring Book: Premium Spirograph Styles Colouring Books for Adult and Kid! Independently Published, 2021.

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Publishing, Skellee. Colouring and Brain Teaser Activities for Kids Ages 6-8 8-10 UK Version - Secret Codes - I Spy - Sudoku - Mazes - Odd One Out - Word Search - Scramble: An Activity Book Packed with over 100 Puzzles. Independently Published, 2020.

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Book chapters on the topic "Odd colouring"

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"colouring | coloring, adj." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/1128518023.

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"colouring | coloring, n." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/4746329854.

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"dead-colouring | dead-coloring, n." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2024. http://dx.doi.org/10.1093/oed/1170837837.

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"colouring in | coloring in, n." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/8578639611.

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"colouring matter | coloring matter, n." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/7325500347.

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"colourish, n." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/7103047581.

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Krishnamurti, Bhadriraju, and Murray B. Emeneau. "Evidence for a Laryngeal *H in Proto-Dravidian." In Comparative Dravidian Linguistics, 307–23. Oxford University PressOxford, 2001. http://dx.doi.org/10.1093/oso/9780198241225.003.0019.

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"colourize | colorize, v." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/6429298936.

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"colourist | colorist, n." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/4842462689.

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"colourant | colorant, n." In Oxford English Dictionary. 3rd ed. Oxford University Press, 2023. http://dx.doi.org/10.1093/oed/5545409665.

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Conference papers on the topic "Odd colouring"

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Bernardi, João Pedro W., Sheila M. De Almeida, and Leandro M. Zatesko. "On Total and Edge-colouring of Proper Circular-arc Graphs." In III Encontro de Teoria da Computação. Sociedade Brasileira de Computação - SBC, 2018. http://dx.doi.org/10.5753/etc.2018.3557.

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Deciding if a graph is Δ-edge-colourable (resp. (Δ + 1)-total colourable), although it is an NP-complete problem for graphs in general, is polynomially solvable for interval graphs of odd (resp. even) maximum degree Δ. An interesting superclass of the proper interval graphs are the proper circular-arc graphs, for which we suspect that Δ-edge-colourability is linear-time decidable. This work presents sufficient conditions for Δ-edge-colourability, (Δ + 1)-total colourability, and (Δ+2)-total colourability of proper circular-arc graphs. Our proofs are constructive and yield polynomial-time algorithms.
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