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11.
While solving a question on the list coloring of planar graphs, Dvo?ák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevi? and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K k, but DP-coloring seems more promising and useful than the Plesnevi?–Vizing reduction. Some properties of the DP-chromatic number χ DP (G) resemble the properties of the list chromatic number χ l (G) but some differ quite a lot. It is always the case that χ DP (G) ≥ χ l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erd?s–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring. 相似文献
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A hypergraph is simple if it has no two edges sharing more than a single vertex. It is s‐list colorable (or s‐choosable) if for any assignment of a list of s colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. We prove that for every positive integer r, there is a function dr(s) such that no r‐uniform simple hypergraph with average degree at least dr(s) is s‐list‐colorable. This extends a similar result for graphs, due to the first author, but does not give as good estimates of dr(s) as are known for d2(s), since our proof only shows that for each fixed r ≥ 2, dr(s) ≤ 2 We use the result to prove that for any finite set of points X in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of X. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 相似文献
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N. V. Zyk S. Z. Vatsadze M. L. Kostochka V. P. Lezina V. G. Vinokurov 《Chemistry of Heterocyclic Compounds》2004,40(1):70-74
Treatment of 1,5- and 1,3-dicarbonyl derivatives of 2,2-dimethyltetrahydropyran with a series of binucleophiles gave condensed pyranopyrazole, pyranothiopyrimidine, pyranoisoxazole, and pyranopyridine systems. 相似文献
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A. V. Kostochka 《Combinatorica》1984,4(4):307-316
The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn vertices, furthermore ifk is large enough then for almost all graphs withn vertices andnk edges. 相似文献
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Families of boxes in the d-dimensional Euclidean space are considered. A simple proof of an upper bound on the size of a minimum transversal in terms of the space dimension and the independence number of the intersection graph of the family is given. The proof uses an idea by Gyárfás and Lehel. Some examples are constructed, showing that in three instances this upper bound is exact. 相似文献
17.
Zoltán Füredi Alexandr Kostochka Ruth Luo Jacques Verstraëte 《Discrete Mathematics》2018,341(5):1253-1263
The Erd?s–Gallai Theorem states that for , any -vertex graph with no cycle of length at least has at most edges. A stronger version of the Erd?s–Gallai Theorem was given by Kopylov: If is a 2-connected -vertex graph with no cycle of length at least , then , where . Furthermore, Kopylov presented the two possible extremal graphs, one with edges and one with edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for odd and all , every -vertex 2-connected graph with no cycle of length at least is a subgraph of one of the two extremal graphs or . The upper bound for here is tight. 相似文献
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Given a fixed multigraph H with V(H) = {h1,…, hm}, we say that a graph G is H‐linked if for every choice of m vertices v1, …, vm in G, there exists a subdivision of H in G such that for every i, vi is the branch vertex representing hi. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008 相似文献