Oleg V. Borodin, Anna O. Ivanova (2013)
Discussiones Mathematicae Graph Theory
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We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40 [...] +1. The bound 2∆−1 is reached at any graph that has two adjacent vertices of degree ∆.
Ghebleh, Mohammad, Kral', Daniel, Norine, Serguei, Thomas, Robin (2006)
The Electronic Journal of Combinatorics [electronic only]
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LeSaulnier, Timothy D., Stocker, Christopher, Wenger, Paul S., West, Douglas B. (2010)
The Electronic Journal of Combinatorics [electronic only]
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Arnfried Kemnitz, Massimiliano Marangio, Margit Voigt (2016)
Discussiones Mathematicae Graph Theory
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Let G = (V,E) be a simple graph and for every edge e ∈ E let L(e) be a set (list) of available colors. The graph G is called L-edge colorable if there is a proper edge coloring c of G with c(e) ∈ L(e) for all e ∈ E. A function f : E → ℕ is called an edge choice function of G and G is said to be f-edge choosable if G is L-edge colorable for every list assignment L with |L(e)| = f(e) for all e ∈ E. Set size(f) = ∑e∈E f(e) and define the sum choice index χ′sc(G) as the minimum of size(f)...
Július Czap, Peter Šugerek, Jaroslav Ivančo (2016)
Discussiones Mathematicae Graph Theory
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An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 𝒦2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 𝒦2(G) for trees, cacti, complete multipartite graphs and graph joins.
Mohar, Bojan, Pisanski, Tomaz (1983)
Publications de l'Institut Mathématique. Nouvelle Série
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Juvan, Martin, Mohar, Bojan, Thomas, Robin (1999)
The Electronic Journal of Combinatorics [electronic only]
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Zhou, Xiao, Nishizeki, Takao (1999)
Journal of Graph Algorithms and Applications
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Li, Jinbo, Liu, Guizhen (2011)
Bulletin of the Malaysian Mathematical Sciences Society. Second Series
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Eric Andrews, Laars Helenius, Daniel Johnston, Jonathon VerWys, Ping Zhang (2014)
Discussiones Mathematicae Graph Theory
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A twin edge k-coloring of a graph G is a proper edge coloring of G with the elements of Zk so that the induced vertex coloring in which the color of a vertex v in G is the sum (in Zk) of the colors of the edges incident with v is a proper vertex coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Among the results presented are formulas for the twin chromatic index of each complete graph and each complete bipartite graph
Albertson, Michael O., Chappell, Glenn G., Kierstead, H.A., Kündgen, André, Ramamurthi, Radhika (2004)
The Electronic Journal of Combinatorics [electronic only]
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Július Czap, Zsolt Tuza (2013)
Discussiones Mathematicae Graph Theory
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An edge coloring of a plane graph G is facially proper if no two faceadjacent edges of G receive the same color. A facial (facially proper) parity edge coloring of a plane graph G is an (facially proper) edge coloring with the property that, for each color c and each face f of G, either an odd number of edges incident with f is colored with c, or color c does not occur on the edges of f. In this paper we deal with the following question: For which integers k does there exist a facial...
Wen-Yao Song, Lian-Ying Miao (2017)
Discussiones Mathematicae Graph Theory
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A strong edge-coloring of a graph is a proper edge-coloring where each color class induces a matching. We denote by 𝜒's(G) the strong chromatic index of G which is the smallest integer k such that G can be strongly edge-colored with k colors. It is known that every planar graph G has a strong edge-coloring with at most 4 Δ(G) + 4 colors [R.J. Faudree, A. Gyárfás, R.H. Schelp and Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205–211]. In this paper, we show that...
Petros A. Petrosyan (2011)
Discussiones Mathematicae Graph Theory
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An edge coloring of a graph G with colors 1,2,...,t is called an interval t-coloring if for each i ∈ {1,2,...,t} there is at least one edge of G colored by i, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable, if there is an integer t ≥ 1 for which G has an interval t-coloring. Let ℜ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if G,H ∈ 𝔑, then the Cartesian product...
William F. Klostermeyer, Gary MacGillivray (2004)
Discussiones Mathematicae Graph Theory
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We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.
Yuehua Bu, Ko-Wei Lih, Weifan Wang (2011)
Discussiones Mathematicae Graph Theory
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An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring o G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ'ₐ(G). We prove that χ'ₐ(G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges whose girth is at least 6. This gives new evidence to a conjecture proposed in [Z. Zhang, L. Liu,...