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Journal articles on the topic 'Proper coloring'

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Author: Grafiati

Published: 25 May 2024

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1

Ma, Baolin, and Chao Yang. "Distinguishing colorings of graphs and their subgraphs." AIMS Mathematics 8, no. 11 (2023): 26561–73. http://dx.doi.org/10.3934/math.20231357.

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<abstract><p>In this paper, several distinguishing colorings of graphs are studied, such as vertex distinguishing proper edge coloring, adjacent vertex distinguishing proper edge coloring, vertex distinguishing proper total coloring, adjacent vertex distinguishing proper total coloring. Finally, some related chromatic numbers are determined, especially the comparison of the correlation chromatic numbers between the original graph and the subgraphs are obtained.</p></abstract>

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2

Goddard, Wayne, and Robert Melville. "Coloring subgraphs with restricted amounts of hues." Open Mathematics 15, no. 1 (September 22, 2017): 1171–80. http://dx.doi.org/10.1515/math-2017-0098.

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Abstract We consider vertex colorings where the number of colors given to specified subgraphs is restricted. In particular, given some fixed graph F and some fixed set A of positive integers, we consider (not necessarily proper) colorings of the vertices of a graph G such that, for every copy of F in G, the number of colors it receives is in A. This generalizes proper colorings, defective coloring, and no-rainbow coloring, inter alia. In this paper we focus on the case that A is a singleton set. In particular, we investigate the colorings where the graph F is a star or is 1-regular.

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3

Chartrand, Gary, James Hallas, and Ping Zhang. "Royal Colorings of Graphs." Ars Combinatoria 156 (July 31, 2023): 51–63. http://dx.doi.org/10.61091/ars156-06.

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For a graph \(G\) and a positive integer \(k\), a royal \(k\)-edge coloring of \(G\) is an assignment of nonempty subsets of the set \(\{1, 2, \ldots, k\}\) to the edges of \(G\) that gives rise to a proper vertex coloring in which the color assigned to each vertex \(v\) is the union of the sets of colors of the edges incident with \(v\). If the resulting vertex coloring is vertex-distinguishing, then the edge coloring is a strong royal \(k\)-coloring. The minimum positive integer \(k\) for which a graph has a strong royal \(k\)-coloring is the strong royal index of the graph. The primary emphasis here is on strong royal colorings of trees.

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Chartrand, Gary, James Hallas, and Ping Zhang. "Royal Colorings of Graphs." Ars Combinatoria 156 (July 31, 2023): 51–63. http://dx.doi.org/10.61091/ars156-6.

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For a graph G and a positive integer k , a royal k -edge coloring of G is an assignment of nonempty subsets of the set { 1 , 2 , … , k } to the edges of G that gives rise to a proper vertex coloring in which the color assigned to each vertex v is the union of the sets of colors of the edges incident with v . If the resulting vertex coloring is vertex-distinguishing, then the edge coloring is a strong royal k coloring. The minimum positive integer k for which a graph has a strong royal k -coloring is the strong royal index of the graph. The primary emphasis here is on strong royal colorings of trees.

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5

Keszegh, Balázs, and Dömötör Pálvölgyi. "Proper Coloring of Geometric Hypergraphs." Discrete & Computational Geometry 62, no. 3 (May 15, 2019): 674–89. http://dx.doi.org/10.1007/s00454-019-00096-9.

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6

Bagheri Gh., Behrooz, and Behnaz Omoomi. "On the simultaneous edge coloring of graphs." Discrete Mathematics, Algorithms and Applications 06, no. 04 (October 10, 2014): 1450049. http://dx.doi.org/10.1142/s1793830914500499.

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A μ-simultaneous edge coloring of graph G is a set of μ proper edge colorings of G with a same color set such that for each vertex, the sets of colors appearing on the edges incident to that vertex are the same in each coloring and no edge receives the same color in any two colorings. The μ-simultaneous edge coloring of bipartite graphs has a close relation with μ-way Latin trades. Mahdian et al. (2000) conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al. (2004) showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 2-simultaneous edge coloring. In this paper, the μ-simultaneous edge coloring of graphs is studied. Moreover, the properties of the extremal counterexample to the above conjecture are investigated. Also, a relation between 2-simultaneous edge coloring of a graph and a cycle double cover with certain properties is shown and using this relation, some results about 2-simultaneous edge colorable graphs are obtained.

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7

Zhou, Yangyang, Dongyang Zhao, Mingyuan Ma, and Jin Xu. "Domination Coloring of Graphs." Mathematics 10, no. 6 (March 21, 2022): 998. http://dx.doi.org/10.3390/math10060998.

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A domination coloring of a graph G is a proper vertex coloring of G, such that each vertex of G dominates at least one color class (possibly its own class), and each color class is dominated by at least one vertex. The minimum number of colors among all domination colorings is called the domination chromatic number, denoted by χdd(G). In this paper, we study the complexity of the k-domination coloring problem by proving its NP-completeness for arbitrary graphs. We give basic results and properties of χdd(G), including the bounds and characterization results, and further research χdd(G) of some special classes of graphs, such as the split graphs, the generalized Petersen graphs, corona products, and edge corona products. Several results on graphs with χdd(G)=χ(G) are presented. Moreover, an application of domination colorings in social networks is proposed.

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8

Li, Minhui, Shumin Zhang, Caiyun Wang, and Chengfu Ye. "The Dominator Edge Coloring of Graphs." Mathematical Problems in Engineering 2021 (October 7, 2021): 1–7. http://dx.doi.org/10.1155/2021/8178992.

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Let G be a simple graph. A dominator edge coloring (DE-coloring) of G is a proper edge coloring in which each edge of G is adjacent to every edge of some color class (possibly its own class). The dominator edge chromatic number (DEC-number) of G is the minimum number of color classes among all dominator edge colorings of G , denoted by χ d ′ G . In this paper, we establish the bounds of the DEC-number of a graph, present the DEC-number of special graphs, and study the relationship of the DEC-number between G and the operations of G .

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9

Kristiana, Arika Indah, Ahmad Aji, Edy Wihardjo, and Deddy Setiawan. "on Graceful Chromatic Number of Vertex amalgamation of Tree Graph Family." CAUCHY: Jurnal Matematika Murni dan Aplikasi 7, no. 3 (October 11, 2022): 432–44. http://dx.doi.org/10.18860/ca.v7i3.16334.

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Proper vertex coloring c of a graph G is a graceful coloring if c is a graceful k-coloring for k∈{1,2,3,…}. Definition graceful k-coloring of a graph G=(V,E) is a proper vertex coloring c:V(G)→{1,2,…,k);k≥2, which induces a proper edge coloring c':E(G)→{1,2,…,k-1} defined c'(uv)=|c(u)-c(v)|. The minimum vertex coloring from graph G can be colored with graceful coloring called a graceful chromatic number with notation χg (G). In this paper, we will investigate the graceful chromatic number of vertex amalgamation of tree graph family with some graph is path graph, centipede graph, broom and E graph.

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10

Sagan, Bruce, and Vincent Vatter. "Bijective Proofs of Proper Coloring Theorems." American Mathematical Monthly 128, no. 6 (May 28, 2021): 483–99. http://dx.doi.org/10.1080/00029890.2021.1901460.

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11

Naduvath, Sudev. "Equitable coloring parameters of certain graph classes." Discrete Mathematics, Algorithms and Applications 10, no. 03 (June 2018): 1850040. http://dx.doi.org/10.1142/s1793830918500404.

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Coloring the vertices of a graph [Formula: see text] subject to given conditions can be considered as a random experiment and corresponding to this experiment, a discrete random variable [Formula: see text] can be defined as the color of a vertex chosen at random, with respect to the given type of coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. A proper coloring [Formula: see text] of a graph [Formula: see text], which assigns colors to the vertices of [Formula: see text] such that the numbers of vertices in any two color classes differ by at most one, is called an equitable coloring of [Formula: see text]. In this paper, we study two statistical parameters of certain graphs, with respect to their equitable colorings.

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Jose, S., and S. Naduvath. "On equitable near-proper coloring of some derived graph classes." Carpathian Mathematical Publications 14, no. 2 (December 30, 2022): 529–42. http://dx.doi.org/10.15330/cmp.14.2.529-542.

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An equitable near-proper coloring of a graph $G$ is a defective coloring in which the number of vertices in any two color classes differ by at most one and the bad edges obtained is minimised by restricting the number of color classes that can have adjacency among their own elements. This paper investigates the equitable near-proper coloring of some derived graph classes like Mycielski graphs, splitting graphs and shadow graphs.

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13

Zhong, Chuang, and Shuangliang Tian. "Neighbor Sum Distinguishing Edge (Total) Coloring of Generalized Corona Product." Journal of Physics: Conference Series 2381, no. 1 (December 1, 2022): 012031. http://dx.doi.org/10.1088/1742-6596/2381/1/012031.

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Abstract The coloring theory of graphs is an important part of graph theory research. The key problem of the coloring theory of graphs is to determine the coloring number of each kind of coloring. Traditional coloring concepts mainly include proper vertex coloring, proper edge coloring, proper total coloring, and so on. In recent years, scholars at home and abroad have put forward some new coloring concepts, such as neighbor vertex distinguishing edge (total) coloring, and neighbor sum distinguishing edge (total) coloring, based on traditional coloring concepts and by adding other constraints. Some valuable results have been obtained, which further enrich the theory of graph coloring. For a proper [k]-edge coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k]-edge coloring of G. For a proper [k]-total coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k]-total coloring of G . In this paper, the coloring method and coloring index are determined by the process of induction and deduction and the construction of the dyeing method, and then the rationality of the method is verified by inverse proof and mathematical induction. If G is a simple graph with the order n ≥ 5 , and hn = (Hi ) i∈{1,2,…,n} is a sequence of disjoint simple graphs, where every Hi is a simple graph with the order m ≥ 7 . In this paper, we study the neighbor sum distinguishing edge(total) coloring of the generalized corona product G○hn of G and hn . The results are as follows: (1) If G is a path with order n , hn = (Hi ) i∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have χ Σ ′ ( G ∘ h n ) = m + 3 (2) If G is a path with order n , hn = (Hi ) i∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have χ Σ ′ ′ ( G ∘ h n ) = m + 4 Due to the late development of neighbor sum distinguishing edge (total) coloring of graphs, the related research results are relatively few. By studying the operation graph of a basic simple graph, we can provide the research basis and reference idea for the corresponding coloring of the general graph class. Therefore, it is of theoretical value to study the neighbor sum distinguishing edge (total) coloring problem of generalized corona products of graphs.

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14

Ghazaryan, Aghasi B., and Petros A. Petrosyan. "ON THE PALETTE INDEX OF GRAPHS HAVING A SPANNING STAR." Proceedings of the YSU A: Physical and Mathematical Sciences 56, no. 3 (259) (October 17, 2022): 85–96. http://dx.doi.org/10.46991/pysu:a/2022.56.3.085.

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A proper edge coloring of a graph $G$ is a mapping $\alpha:E(G)\longrightarrow \mathbb{N}$ such that $\alpha(e)\not=\alpha(e')$ for every pair of adjacent edges $e$ and $e'$ in $G$. In a proper edge coloring of a graph $G$, the palette of a vertex $v \in V(G)$ is the set of colors assigned to the edges incident to $v$. The palette index of $G$ is the minimum number of distinct palettes occurring in $G$ among all proper edge colorings of $G$. A graph $G$ has a spanning star, if it has a spanning subgraph which is a star. In this paper, we consider the palette index of graphs having a spanning star. In particular, we give sharp upper and lower bounds on the palette index of these graphs. We also provide some upper and lower bounds on the palette index of the complete split and threshold graphs.

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15

Keszegh, Balázs. "Coloring Intersection Hypergraphs of Pseudo-Disks." Discrete & Computational Geometry 64, no. 3 (October 30, 2019): 942–64. http://dx.doi.org/10.1007/s00454-019-00142-6.

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Abstract We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with four colors and a conflict-free coloring with $$O(\log n)$$ O ( log n ) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with $$O(\log n)$$ O ( log n ) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.

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16

DONG, AIJUN, and GUANGHUI WANG. "NEIGHBOR SUM DISTINGUISHING COLORING OF SOME GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 04 (December 2012): 1250047. http://dx.doi.org/10.1142/s1793830912500474.

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A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.

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17

Zhao, Xiao, and Sheng Chen. "Proper 2-coloring game on some trees." Theoretical Computer Science 778 (July 2019): 1–18. http://dx.doi.org/10.1016/j.tcs.2019.01.021.

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18

Liang, Zuosong, and Huandi Wei. "A Linear-Time Algorithm for 4-Coloring Some Classes of Planar Graphs." Computational Intelligence and Neuroscience 2021 (October 5, 2021): 1–5. http://dx.doi.org/10.1155/2021/7667656.

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Every graph G = V , E considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e ∈ E of G with an unordered pair of vertices of G which are called the ends of the edge e . A graph is said to be a planar graph if it can be drawn in the plane so that its edges intersect only at their ends. A proper k -vertex-coloring of a graph G = V , E is a mapping c : V ⟶ S ( S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar graph has a proper vertex-coloring with four colors. However, the current known proof for the Four Color Theorem is computer assisted. In addition, the correctness of the proof is still lengthy and complicated. In 2010, a simple O n 2 time algorithm was provided to 4-color a 3-colorable planar graph. In this paper, we give an improved linear-time algorithm to either output a proper 4-coloring of G or conclude that G is not 3-colorable when an arbitrary planar graph G is given. Using this algorithm, we can get the proper 4-colorings of 3-colorable planar graphs, planar graphs with maximum degree at most five, and claw-free planar graphs.

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Setyorini, Miftakhul Jannah, and I. Ketut Budayasa. "Bilangan Kromatik-b Graf Sentral, Tengah, Total dari Sebuah Bintang." MATHunesa: Jurnal Ilmiah Matematika 9, no. 1 (January 26, 2021): 36–42. http://dx.doi.org/10.26740/mathunesa.v9n1.p36-42.

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Let G be a graph. A proper k-coloring of G is coloring all vertices of G with k colors such that every two adjacent vertices are assigned different colors. The minimum value of k for which a proper k-coloring of G exist is called the chromatic number of G. A b-coloring of G is a proper k-coloring of G such that each color class has a representative that is adjacent to at least one vertex in each of the other color classes. The largest positive integer k such that there is a b-coloring of G is called the b-chromatic number of G, denoted . In this article, we establish the b-chromatic number of the central graph on the star graph , the b-chromatic number of the middle graph on the star graph and the b-chromatic number of total graph on the star graph . Keywords : Chromatic Number; B-chromatic Number; Star Graf

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20

Indah Kristiana, Arika, Anzori Anzori, Robiatul Adawiyah, Slamin Slamin, and Ermita Rizki Albirri. "Bilangan Kromatik Graceful Pada Keluarga Graf Grid." Jurnal Axioma : Jurnal Matematika dan Pembelajaran 8, no. 2 (April 11, 2023): 144–55. http://dx.doi.org/10.56013/axi.v8i2.1544.

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All graph in this paper be a connected and simple graph. Let c:V(G)→{1,2,…,k} is a proper vertex coloring where k ≥ 2 which induces a proper edge coloring c':E(G)→{1,2,…,k} define by c' (uv)=|c(u)-c(v)|, where uv in E(G) is called graceful k-coloring. A vertex coloring c of graph G is a graceful coloring if c is a graceful k-coloring for some k∈ N. The minimum k for which a graph G is a graceful chromatic number denoted by χ_g (G). In this paper, we will investigate the establish exact value of graceful chromatic number on grid graph family namely H graph 〖(H〗_n) for n≥2 and grid graph for m,n≥3. Keywords: graceful coloring, graceful chromatic number, grid graph family

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21

Li, Jingwen, Tengyun Hu, and Fei Wen. "The algorithm for adjacent vertex distinguishing proper edge coloring of graphs." Discrete Mathematics, Algorithms and Applications 07, no. 04 (December 2015): 1550044. http://dx.doi.org/10.1142/s1793830915500445.

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An adjacent vertex distinguishing proper edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that no pair of adjacent vertices meet the same set of colors. The minimum number of colors is called adjacent vertex distinguishing proper edge chromatic number of [Formula: see text]. In this paper, we present a new heuristic intelligent algorithm to calculate the adjacent vertex distinguishing proper edge chromatic number of graphs. To be exact, the algorithm establishes two objective subfunctions and a main objective function to find its optimal solutions by the conditions of adjacent vertex distinguishing proper edge coloring. Moreover, we test and analyze its feasibility, and the test results show that this algorithm can rapidly and efficiently calculate the adjacent vertex distinguishing proper edge chromatic number of graphs with fixed order, and its time complexity is less than [Formula: see text].

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22

Kavitha, K., N. G. David, and N. Selvi. "Split and Non-Split Dominator Chromatic Numbers and Related Parameters." Mapana - Journal of Sciences 10, no. 1 (June 30, 2011): 52–62. http://dx.doi.org/10.12723/mjs.18.5.

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A proper graph coloring is defined as coloring the nodes of a graph with the minimum number of colors without any two adjacent nodes having the same color. Dominator coloring of G is a proper coloring in which every vertex of G dominates every vertex of at least one color class. In this paper, new parameters, namely strong split and non-split dominator chromatic numbers and block, cycle, path non-split dominator chromatic numbers are introduced. These parameters are obtained for different classes of graphs and also interesting results are established.

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Fornasiero, Federico, and Sudev Naduvath. "On J-colorability of certain derived graph classes." Acta Universitatis Sapientiae, Informatica 11, no. 2 (December 1, 2019): 159–73. http://dx.doi.org/10.2478/ausi-2019-0011.

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Abstract A vertex v of a given graph G is said to be in a rainbow neighbourhood of G, with respect to a proper coloring C of G, if the closed neighbourhood N[v] of the vertex v consists of at least one vertex from every color class of G with respect to C. A maximal proper coloring of a graph G is a J-coloring of G such that every vertex of G belongs to a rainbow neighbourhood of G. In this paper, we study certain parameters related to J-coloring of certain Mycielski-type graphs.

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24

Cai, Jin, Shuangliang Tian, and Lizhen Peng. "On star and acyclic coloring of generalized lexicographic product of graphs." AIMS Mathematics 7, no. 8 (2022): 14270–81. http://dx.doi.org/10.3934/math.2022786.

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<abstract><p>A $ star \; coloring $ of a graph $ G $ is a proper vertex coloring of $ G $ such that any path of length 3 in $ G $ is not bicolored. The $ star \; chromatic \; number $ $ \chi_s(G) $ of $ G $ is the smallest integer $ k $ for which $ G $ admits a star coloring with $ k $ colors. A $ acyclic \; coloring $ of $ G $ is a proper coloring of $ G $ such that any cycle in $ G $ is not bicolored. The $ acyclic \; chromatic \; number $ of $ G $, denoted by $ a(G) $, is the minimum number of colors needed to acyclically color $ G $. In this paper, we present upper bound for the star and acyclic chromatic numbers of the generalized lexicographic product $ G[h_n] $ of graph $ G $ and disjoint graph sequence $ h_n $, where $ G $ exists a $ k- $colorful neighbor star coloring or $ k- $colorful neighbor acyclic coloring. In addition, the upper bounds are tight.</p></abstract>

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25

Gügümcü, Neslihan, and Sam Nelson. "Biquandle coloring invariants of knotoids." Journal of Knot Theory and Its Ramifications 28, no. 04 (April 2019): 1950029. http://dx.doi.org/10.1142/s0218216519500299.

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In this paper, we consider biquandle colorings for knotoids in [Formula: see text] or [Formula: see text], and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle counting invariant by using a matrix constructed by utilizing the orientation a knotoid diagram is endowed with. We generalize Niebrzydowski’s biquandle longitude invariant for virtual long knots to obtain new invariants for knotoids. We show that biquandle invariants can detect mirror images of knotoids and show that our enhancements are proper in the sense that knotoids which are not distinguished by the counting invariant are distinguished by our enhancements.

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Aisyah, Siti, Ridho Alfarisi, Rafiantika M. Prihandini, Arika Indah Kristiana, and Ratna Dwi Christyanti. "On The Local Edge Antimagic Coloring of Corona Product of Path and Cycle." CAUCHY 6, no. 1 (December 4, 2019): 40. http://dx.doi.org/10.18860/ca.v6i1.8054.

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<p>Let be a nontrivial and connected graph of vertex set and edge set . A bijection is called a local edge antimagic labeling if for any two adjacent edges and , where for . Thus, the local edge antimagic labeling induces a proper edge coloring of G if each edge e assigned the color . The color of each an edge <em>e</em> = <em>uv</em> is assigned bywhich is defined by the sum of label both and vertices and . The local edge antimagic chromatic number, denoted by is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of . In our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle.</p><p><strong>Keywords:</strong> Local antimagic; edge coloring; corona product; path; cycle.</p>

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Asy’ari, M. L., Dafik, I. H. Agustin, R. Nisviasari, and R. Adawiyah. "On graceful chromatic number of some graphs." Journal of Physics: Conference Series 2157, no. 1 (January 1, 2022): 012013. http://dx.doi.org/10.1088/1742-6596/2157/1/012013.

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Abstract We examine that all graphs in this paper are limited, simple and connected. A graceful k-coloring of a graph is a proper vertex coloring f 1 : V (G) → {1, 2,…, k} where k ≥ 2 which induces a proper edge coloring f 2 : E (G) → {1, 2,…, k − 1} characterized by f 2(uυ) = |f 1(u) — f 2 (υ)|. Nethermost k for which a graph G has a graceful k-coloring is named a graceful chromatic number of a graph G, denoted by χg (G). In our research, we will obtain the exact value of the graceful chromatic number of some graphs.

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Ma, Chun Yan, Xiang En Chen, Fang Yang, and Bing Yao. "On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs." Applied Mechanics and Materials 333-335 (July 2013): 1452–55. http://dx.doi.org/10.4028/www.scientific.net/amm.333-335.1452.

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A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

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Panda, B. S., and Arti Pandey. "On the dominator coloring in proper interval graphs and block graphs." Discrete Mathematics, Algorithms and Applications 07, no. 04 (December 2015): 1550043. http://dx.doi.org/10.1142/s1793830915500433.

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In a graph [Formula: see text], a vertex [Formula: see text] dominates a vertex [Formula: see text] if either [Formula: see text] or [Formula: see text] is adjacent to [Formula: see text]. A subset of vertex set [Formula: see text] that dominates all the vertices of [Formula: see text] is called a dominating set of graph [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text] and is denoted by [Formula: see text]. A proper coloring of a graph [Formula: see text] is an assignment of colors to the vertices of [Formula: see text] such that any two adjacent vertices get different colors. The minimum number of colors required for a proper coloring of [Formula: see text] is called the chromatic number of [Formula: see text] and is denoted by [Formula: see text]. A dominator coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of [Formula: see text] is called the dominator chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we study the dominator chromatic number for the proper interval graphs and block graphs. We show that every proper interval graph [Formula: see text] satisfies [Formula: see text], and these bounds are sharp. For a block graph [Formula: see text], where one of the end block is of maximum size, we show that [Formula: see text]. We also characterize the block graphs with an end block of maximum size and attaining the lower bound.

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30

Qin, Zhongmei, and Junxue Zhang. "Extremal stretch of proper-walk coloring of graphs." Applied Mathematics and Computation 405 (September 2021): 126240. http://dx.doi.org/10.1016/j.amc.2021.126240.

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31

Kumar, Dr J. Suresh. "Unique Minimal Proper Roman Coloring of a Graph." International Journal for Research in Applied Science and Engineering Technology 8, no. 4 (April 30, 2020): 89–91. http://dx.doi.org/10.22214/ijraset.2020.4015.

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32

Pillai, Preethi K. "Proper Roman Coloring of some Cycle related Graphs." International Journal for Research in Applied Science and Engineering Technology 8, no. 5 (May 31, 2020): 2080–84. http://dx.doi.org/10.22214/ijraset.2020.5341.

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33

Diwan, Ajit, Soumitra Pal, and Abhiram Ranade. "Fragmented coloring of proper interval and split graphs." Discrete Applied Mathematics 193 (October 2015): 110–18. http://dx.doi.org/10.1016/j.dam.2015.04.014.

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34

Chandran, L. Sunil, Sajal K. Das, Pavol Hell, Sajith Padinhatteeri, and Raji R. Pillai. "Template-driven rainbow coloring of proper interval graphs." Discrete Applied Mathematics 328 (March 2023): 97–107. http://dx.doi.org/10.1016/j.dam.2022.12.009.

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35

Maulana, Nur Ridwan, Kristiana Wijaya, and Kiswara Agung Santoso. "POLINOMIAL KROMATIK PADA GRAF KIPAS." Majalah Ilmiah Matematika dan Statistika 18, no. 2 (September 3, 2018): 55. http://dx.doi.org/10.19184/mims.v18i2.17248.

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A chromatic polynomial of a graph G is a special function that describes the number of ways we can achieve a proper coloring on the vertices of G given k colors. In this paper, we determine a chromatic polynomial of a fan graph. Keywords: Proper coloring, chromatic polynomial, fan graph.

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36

Tolentino, J. D., M. A. C. Tolentino, and E. B. Bernales. "On twin edge mean colorings of graphs." Journal of Physics: Conference Series 2157, no. 1 (January 1, 2022): 012005. http://dx.doi.org/10.1088/1742-6596/2157/1/012005.

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Abstract Let k ≥ 2 be an integer and G be a connected graph of order at least 3. In this paper, we introduce a new neighbor-distinguishing coloring called twin edge mean coloring. A proper edge coloring of G that uses colors from ℕ k = {0,1,…, k − 1} is called a twin k-edge mean coloring of G if it induces a proper vertex coloring of G such that the color of each vertex υ of G is the average of the colors of the edges incident with υ, and is an integer. The minimum k for which G has a twin k-edge mean coloring is called the twin chromatic mean index of G and is denoted by χ t m ′ ( G ) . First, we establish lower and upper bounds for χ t m ′ ( G ) under general or more specific assumptions. Then we determine the twin chromatic mean indices of paths, cycles, and stars.

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37

K.S, Kanzul Fathima, and Jahir Hussain R. "An Introduction to Fuzzy Edge Coloring." JOURNAL OF ADVANCES IN MATHEMATICS 11, no. 10 (January 28, 2016): 5742–48. http://dx.doi.org/10.24297/jam.v11i10.801.

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In this paper, a new concept of fuzzy edge coloring is introduced. The fuzzy edge coloring is an assignment of colors to edges of a fuzzy graph G. It is proper if no two strong adjacent edges of G will receive the same color. Fuzzy edge chromatic number of G is least positive integer for which G has a proper fuzzy edge coloring. In this paper, the fuzzy edge chromatic number of different classes of fuzzy graphs and the fuzzy edge chromatic number of fuzzy line graphs are found. Isochromatic fuzzy graph is also defined.

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Samanta, Sovan, Jeong Gon Lee, Usman Naseem, Shah Khalid Khan, and Kousik Das. "Concepts on Coloring of Cluster Hypergraphs with Application." Mathematical Problems in Engineering 2020 (August 11, 2020): 1–10. http://dx.doi.org/10.1155/2020/3705156.

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Coloring of graph theory is widely used in different fields like the map coloring, traffic light problems, etc. Hypergraphs are an extension of graph theory where edges contain single or multiple vertices. This study analyzes cluster hypergraphs where cluster vertices too contain simple vertices. Coloring of cluster networks where composite/cluster vertices exist is done using the concept of coloring of cluster hypergraphs. Proper coloring and strong coloring of cluster hypergraphs have been defined. Along with these, local coloring in cluster hypergraphs is also provided. Such a cluster network, COVID19 affected network, is assumed and colored to visualize the affected regions properly.

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Cho, Karina, and Sam Nelson. "Quandle coloring quivers." Journal of Knot Theory and Its Ramifications 28, no. 01 (January 2019): 1950001. http://dx.doi.org/10.1142/s0218216519500019.

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We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.

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Bendali-Braham, Amel, Noureddine Ikhlef-Eschouf, and Mostafa Blidia. "Some results on the b-chromatic number in complementary prism graphs." RAIRO - Operations Research 53, no. 4 (July 29, 2019): 1187–95. http://dx.doi.org/10.1051/ro/2018054.

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A b-coloring of a graph G is a proper coloring of G with k colors such that each color class has a vertex that is adjacent to at least one vertex of every other color classes. The b-chromatic number is the largest integer k for which G has a b-coloring with k colors. In this paper, we present some results on b-coloring in complementary prism graphs.

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S. K. Vaidya and Rakhimol V. Isaac. "The b-chromatic number of some degree splitting graphs." Malaya Journal of Matematik 2, no. 03 (July 1, 2014): 249–53. http://dx.doi.org/10.26637/mjm203/010.

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A $b$-coloring of a graph $G$ is a variant of proper coloring in which each color class contains a vertex that has a neighbor in all the other color classes. We investigate some results on $b$-coloring in the context of degree splitting graph of $P_n, B_{n, n}, S_n$ and $G_n$.

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Kaveh, Kamran, Alex McAvoy, Krishnendu Chatterjee, and Martin A. Nowak. "The Moran process on 2-chromatic graphs." PLOS Computational Biology 16, no. 11 (November 5, 2020): e1008402. http://dx.doi.org/10.1371/journal.pcbi.1008402.

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Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where “properly colored” means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring.

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Arumugam, S., and K. Raja Chandrasekar. "Linear time algorithm for dominator chromatic number of trestled graphs." Discrete Mathematics, Algorithms and Applications 11, no. 06 (December 2019): 1950066. http://dx.doi.org/10.1142/s1793830919500666.

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A dominator coloring (respectively, total dominator coloring) of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] such that each closed neighborhood (respectively, open neighborhood) of every vertex of [Formula: see text] contains a color class of [Formula: see text] The minimum number of colors required for a dominator coloring (respectively, total dominator coloring) of [Formula: see text] is called the dominator chromatic number (respectively, total dominator chromatic number) of [Formula: see text] and is denoted by [Formula: see text] (respectively, [Formula: see text]). In this paper, we prove that the dominator coloring problem and the total dominator coloring problem are solvable in linear time for trestled graphs.

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Naduvath, Sudev, and Johan Kok. "J-coloring of graph operations." Acta Universitatis Sapientiae, Informatica 11, no. 1 (August 1, 2019): 95–108. http://dx.doi.org/10.2478/ausi-2019-0007.

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Abstract A vertex v of a given graph is said to be in a rainbow neighbourhood of G if every color class of G consists of at least one vertex from the closed neighbourhood N[v]. A maximal proper coloring of a graph G is a J-coloring if and only if every vertex of G belongs to a rainbow neighbourhood of G. In general all graphs need not have a J-coloring, even though they admit a chromatic coloring. In this paper, we characterise graphs which admit a J-coloring. We also discuss some preliminary results in respect of certain graph operations which admit a J-coloring under certain conditions.

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Hamiez, Jean-Philippe, Jin-Kao Hao, and Fred W. Glover. "A Study of Tabu Search for Coloring Random 3-Colorable Graphs Around the Phase Transition." International Journal of Applied Metaheuristic Computing 1, no. 4 (October 2010): 1–24. http://dx.doi.org/10.4018/jamc.2010100101.

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The authors present an experimental investigation of tabu search (TS) to solve the 3-coloring problem (3-COL). Computational results reveal that a basic TS algorithm is able to find proper 3-colorings for random 3-colorable graphs with up to 11000 vertices and beyond when instances follow the uniform or equipartite well-known models, and up to 1500 vertices for the hardest class of flat graphs. This study also validates and reinforces some existing phase transition thresholds for 3-COL.

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Thamil Selvi.M.S, Franklin, Amutha A, and Antony Mary A. "A Study on Harmonious Coloring of Circulant Networks." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 393. http://dx.doi.org/10.14419/ijet.v7i4.10.20945.

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Given a simple graph , a harmonious coloring of is the proper vertex coloring such that each pair of colors seems to appears together on at most one edge. The harmonious chromatic number of , denoted by is the minimal number of colors in a harmonious coloring of . In this paper we have determined the harmonious chromatic number of some classes of Circulant Networks.

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Ge, Wei, and Jun Yue. "Total Dominator Colorings of P 4 -reducible and P 4 -tidy Graphs." Ars Combinatoria 157 (December 31, 2023): 81–88. http://dx.doi.org/10.61091/ars157-08.

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A total dominator coloring of G without isolated vertex is a proper coloring of the vertices of G in which each vertex of G is adjacent to every vertex of some color class. The total dominator chromatic number χ t d ( G ) of G is the minimum number of colors among all total dominator coloring of G . In this paper, we will give the polynomial time algorithms to computing the total dominator coloring number for P 4 -reducible and P 4 -tidy graphs.

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Naveen, J. "Injective Edge Coloring of Cubic Graphs." Journal of Mathematical Sciences & Computational Mathematics 3, no. 1 (October 4, 2021): 26–49. http://dx.doi.org/10.15864/jmscm.3103.

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Three edges e1, e2 and e3 in a graph G are consecutive if they form a cycle of length 3 or a path in this order. A k-injective edge-coloring of a graph G is an edge-coloring of G, (not necessarily proper), such that if edges e1, e2, e3 are consecutive, then e1 and e3 receive distinct colors. The minimum k for which G has a k-injective edge-coloring is called the injective edge-coloring number, denoted by χ′i(G). In this paper, injective edge-coloring numbers of H-graph and generalized H-graph are determined.

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Yegnanarayanan, Venkataraman, Gayathri Yegnanarayanan, and Marius Balas. "On Coloring Catalan Number Distance Graphs and Interference Graphs." Symmetry 10, no. 10 (October 9, 2018): 468. http://dx.doi.org/10.3390/sym10100468.

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A vertex coloring of a graph G is a mapping that allots colors to the vertices of G. Such a coloring is said to be a proper vertex coloring if two vertices joined by an edge receive different colors. The chromatic number χ ( G ) is the least number of colors used in a proper vertex coloring. In this paper, we compute the χ of certain distance graphs whose distance set elements are (a) a finite set of Catalan numbers, (b) a finite set of generalized Catalan numbers, (c) a finite set of Hankel transform of a transformed sequence of Catalan numbers. Then while discussing the importance of minimizing interference in wireless networks, we probe how a vertex coloring problem is related to minimizing vertex collisions and signal clashes of the associated interference graph. Then when investigating the χ of certain G ( V , D ) and graphs with interference, we also compute certain lower and upper bound for χ of any given simple graph in terms of the average degree and Laplacian operator. Besides obtaining some interesting results we also raised some open problems.

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R.R, Aaresh, Venkatachalam M, and Deepa T. "ON DYNAMIC COLORING OF WEB GRAPH." Kongunadu Research Journal 5, no. 2 (December 30, 2018): 11–15. http://dx.doi.org/10.26524/krj263.

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Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

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