منابع مشابه
List-color-critical graphs on a fixed surface
A k-list-assignment for a graph G assigns to each vertex v of G a list L(v) of admissible colors , where |L(v)| ≥ k. A graph is k-list-colorable (or k-choosable) if it can be properly colored from the lists for every k-listassignment. We prove the following conjecture posed by Thomassen in 1994: “There are only finitely many listcolor-critical graphs with all lists of cardinality at least 5 on ...
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Let K be a class of graphs. Then, K is said to have a finite duality if there exists a pair (F , U), where U ∈ K and F is a finite set of graphs, such that for any graph G in K we have G ≤ U if and only if F 6≤ G for all F ∈ F (“ ≤ ” is the homomorphism order). We prove that the class of planar graphs has no finite duality except for two trivial cases. We also prove that a 5-colorable toroidal ...
متن کاملOn complete subgraphs of color-critical graphs
A graph G is called k-critical if x(G) = k and x(G e) -C x(G) for each edge e of G, where x denotes the chromatic number. T. Gallai conjectured that every k-critical graph of order n contains at most n complete (k l)-subgraphs. In 1987, Stiebitz proved Gallai’s conjecture in the case k = 4, and in 1992 Abbott and Zhou proved Gallai’s conjecture for all k > 5. In their paper, Abbott and Zhou ask...
متن کاملLecture notes on sparse color-critical graphs
This text together with the attached paper [8] surveys results on color-critical graphs, with emphasis on sparse ones. The first two sections discuss the important contributions by Dirac and Gallai and present proofs of some remarkable results of them. The next two sections discuss the later progress and a number of applications of the recent results. We also use [8] for description of some app...
متن کاملLong Cycles in Graphs on a Fixed Surface
We prove that there exists a function a : N0 × R+ → N such that (i) If G is a 4-connected graph embedded on a surface of Euler genus g such that the face-width of G is at least a(g, ε), then G can be covered by two cycles each of which has length at least (1− ε)n. We apply this to derive lower bounds for the length of a longest cycle in a graph G on any fixed surface. Specifically, there exist ...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 1997
ISSN: 0095-8956
DOI: 10.1006/jctb.1996.1722