共查询到20条相似文献，搜索用时 15 毫秒
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《Journal of Graph Theory》2018,88(3):521-546
Correspondence coloring, or DP‐coloring, is a generalization of list coloring introduced recently by Dvořák and Postle [11]. In this article, we establish a version of Dirac's theorem on the minimum number of edges in critical graphs [9] in the framework of DP‐colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz [15] on classifying list‐critical graphs that satisfy Dirac's bound with equality. 相似文献
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A graph G is called ‐choosable if for any list assignment L that assigns to each vertex v a set of a permissible colors, there is a b‐tuple L‐coloring of G . An (a , 1)‐choosable graph is also called a‐choosable. In the pioneering article on list coloring of graphs by Erd?s et al. 2 , 2‐choosable graphs are characterized. Confirming a special case of a conjecture in 2 , Tuza and Voigt 3 proved that 2‐choosable graphs are ‐choosable for any positive integer m . On the other hand, Voigt 6 proved that if m is an odd integer, then these are the only ‐choosable graphs; however, when m is even, there are ‐choosable graphs that are not 2‐choosable. A graph is called 3‐choosable‐critical if it is not 2‐choosable, but all its proper subgraphs are 2‐choosable. Voigt conjectured that for every positive integer m , all bipartite 3‐choosable‐critical graphs are ‐choosable. In this article, we determine which 3‐choosable‐critical graphs are (4, 2)‐choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer k such that for any positive integer m , every bipartite 3‐choosable‐critical graph is ‐choosable. Moving beyond 3‐choosable‐critical graphs, we present an infinite family of non‐3‐choosable‐critical graphs that have been shown by computer analysis to be (4, 2)‐choosable. This shows that the family of all (4, 2)‐choosable graphs has rich structure. 相似文献
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Let be a function on the vertex set of the graph . The graph G is f‐choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number, , is the minimum of , over all functions f such that G is f‐choosable. It is known (Alon, Surveys in Combinatorics, 1993 (Keele), London Mathematical Society Lecture Note Series, Vol. 187, Cambridge University Press, Cambridge, 1993, pp. 1–33, Random Struct Algor 16 (2000), 364–368) that if G has average degree d, then the usual choice number is at least , so they grow simultaneously. In this article, we show that can be bounded while the minimum degree . Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph . 相似文献
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A graph is a k‐critical graph if G is not ‐colorable but every proper subgraph of G is ‐colorable. In this article, we construct a family of 4‐critical planar graphs with n vertices and edges. As a consequence, this improves the bound for the maximum edge density attained by Abbott and Zhou. We conjecture that this is the largest edge density for a 4‐critical planar graph. 相似文献
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This paper proves the following result. Assume
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Joan P. Hutchinson 《Journal of Graph Theory》2008,59(1):59-74
We prove that a 2‐connected, outerplanar bipartite graph (respectively, outerplanar near‐triangulation) with a list of colors L (v ) for each vertex v such that (resp., ) can be L‐list‐colored (except when the graph is K3 with identical 2‐lists). These results are best possible for each condition in the hypotheses and bounds. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 59–74, 2008 相似文献
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A 2‐assignment on a graph G = (V,E) is a collection of pairs L(v) of allowed colors specified for all vertices v ∈V. The graph G (with at least one edge) is said to have oriented choice number 2 if it admits an orientation which satisfies the following property: For every 2‐assignment there exists a choice c(v)∈L(v) for all v ∈V such that (i) if c(v) = c(w), then vw ∉ E, and (ii) for every ordered pair (a,b) of colors, if some edge oriented from color a to color b occurs, then no edge is oriented from color b to color a. In this paper we characterize the following subclasses of graphs of oriented choice number 2: matchings; connected graphs; graphs containing at least one cycle. In particular, the first result (which implies that the matching with 11 edges has oriented choice number 2) proves a conjecture of Sali and Simonyi. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 217–229, 2001 相似文献
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Julia Böttcher Yoshiharu Kohayakawa Aldo Procacci 《Random Structures and Algorithms》2012,40(4):425-436
Let G be a graph on n vertices with maximum degree Δ. We use the Lovász local lemma to show the following two results about colourings χ of the edges of the complete graph Kn. If for each vertex v of Kn the colouring χ assigns each colour to at most (n ‐ 2)/(22.4Δ2) edges emanating from v, then there is a copy of G in Kn which is properly edge‐coloured by χ. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409–433, 2003]. On the other hand, if χ assigns each colour to at most n/(51Δ2) edges of Kn, then there is a copy of G in Kn such that each edge of G receives a different colour from χ. This proves a conjecture of Frieze and Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a framework developed by Lu and Székely [Electron. J. Comb. 14(1), R63, 2007] for applying the local lemma to random injections. In order to improve the constants in our results we use a version of the local lemma due to Bissacot, Fernández, Procacci, and Scoppola [preprint, arXiv:0910.1824]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 425–436, 2012 相似文献
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A generalization of the circular chromatic number to hypergraphs is discussed. In particular, it is indicated how the basic theory, and five equivalent formulations of the circular chromatic number of graphs, can be carried over to hypergraphs with essentially the same proofs. 相似文献
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Konrad K. Dabrowski François Dross Matthew Johnson Daniël Paulusma 《Journal of Graph Theory》2019,92(4):377-393
A colouring of a graph is a function such that for every . A -regular list assignment of is a function with domain such that for every , is a subset of of size . A colouring of respects a -regular list assignment of if for every . A graph is -choosable if for every -regular list assignment of , there exists a colouring of that respects . We may also ask if for a given -regular list assignment of a given graph , there exists a colouring of that respects . This yields the -Regular List Colouring problem. For , we determine a family of classes of planar graphs, such that either -Regular List Colouring is -complete for instances with , or every is -choosable. By using known examples of non--choosable and non--choosable graphs, this enables us to classify the complexity of -Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs, and planar graphs with no -cycles and no -cycles. We also classify the complexity of -Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree. 相似文献
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Let G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W?V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K4‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009 相似文献
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We prove a conjecture of Ohba that says that every graph G on at most vertices satisfies . 相似文献
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In 1968, Vizing proposed the following conjecture: If G=(V,E) is a Δ-critical graph of order n and size m, then . This conjecture has been verified for the cases of Δ≤5. In this paper, we prove that when Δ=4. It improves the known bound for Δ=4 when n>6. 相似文献
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Annegret Wagler 《Journal of Graph Theory》1999,32(4):394-404
A perfect graph is critical, if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We relate critically perfect graphs to well-known classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 394–404, 1999 相似文献
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In a recent seminal work, Kostochka and Yancey proved that for every 4‐critical graph G. In this article, we prove that for every 4‐critical graph G with girth at least five. When combined with another result of the second author, the improvement on the constant term leads to a corollary that there exist such that for every 4‐critical graph G with girth at least five. Moreover, it provides a unified and shorter proof of both a result of Thomassen and a result of Thomas and Walls without invoking any topological property, where the former proves that every graph with girth five embeddable in the projective plane or torus is 3‐colorable, and the latter proves the same for the Klein bottle. 相似文献
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Matthias Kriesell 《Journal of Graph Theory》2001,36(1):35-51
A noncomplete graph G is called an (n, k)‐graph if it is n‐connected and G − X is not (n − |X| + 1)‐connected for any X ⊆ V(G) with |X| ≤ k. Mader conjectured that for k ≥ 3 the graph K2k + 2 − (1‐factor) is the unique (2k, k)‐graph. We settle this conjecture for strongly regular graphs, for edge transitive graphs, and for vertex transitive graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 35–51, 2001 相似文献