共查询到20条相似文献,搜索用时 29 毫秒
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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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Recently, Mubayi and Wang showed that for and , the number of -vertex -graphs that do not contain any loose cycle of length is at most . We improve this bound to . 相似文献
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Benjamin Braun Hugo Corrales Scott Corry Luis David García Puente Darren Glass Nathan Kaplan Jeremy L. Martin Gregg Musiker Carlos E. Valencia 《Discrete Mathematics》2018,341(10):2949-2963
Let be a finite, connected graph. An arithmetical structure on is a pair of positive integer vectors such that , where is the adjacency matrix of . We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices ). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles. 相似文献
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Füredi conjectured that the Boolean lattice can be partitioned into chains such that the size of any two differs in at most one. In this paper, we prove that there is an absolute constant with the following property: for every , if is sufficiently large, the Boolean lattice has a chain partition into chains, each of them of size between and .We deduce this result by looking at the more general setup of unimodal normalized matching posets. We prove that a unimodal normalized matching poset of width has a chain partition into chains, each of size at most , and it has a chain partition into chains, where each chain has size at least . 相似文献
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This current paper is devoted to the Cauchy problem for higher order dispersive equation u_t+ ?_x~(2n+1)u = ?_x(u?_x~nu) + ?_x~(n-1)(u_x~2), n ≥ 2, n ∈ N~+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H~(-n/2+3/4,-1/(2n))(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H~(s,a)(R) with s -n/2+3/4 and all a∈R. 相似文献
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A straight-line drawing of a planar graph is a closed rectangle-of-influence drawing if for each edge , the closed axis-parallel rectangle with opposite corners and contains no other vertices. We show that each quadrangulation on vertices has a closed rectangle-of-influence drawing on the grid.The algorithm is based on angle labeling and simple face counting in regions. This answers the question of what would be a grid embedding of quadrangulations analogous to Schnyder’s classical algorithm for embedding triangulations and extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden.A further compaction step yields a straight-line drawing of a quadrangulation on the grid. The advantage over other existing algorithms is that it is not necessary to add edges to the quadrangulation to make it -connected. 相似文献
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The first author showed that the list chromatic number of every graph with average degree is at least . We prove that for , every -uniform hypergraph in which at least half of the -vertex subsets are contained in at least edges has list chromatic number at least . When is fixed, this is sharp up to a constant factor. 相似文献
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The Turán number is the maximum number of edges in any -vertex graph that does not contain a subgraph isomorphic to . A wheel is a graph on vertices obtained from a by adding one vertex and making adjacent to all vertices of the . We obtain two exact values for small wheels: Given that is already known, this paper completes the spectrum for all wheels up to 7 vertices. In addition, we present the construction which gives us the lower bound in general case. 相似文献
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A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph with girth at least is packable.The conjecture was proved by Faudree et al. with the additional condition that has at most edges. In this paper, for each integer , we prove that every non-star graph with girth at least and at most edges is packable, where is for every . This implies that the conjecture is true for sufficiently large planar graphs. 相似文献
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The slow-coloring game is played by Lister and Painter on a graph . On each round, Lister marks a nonempty subset of the uncolored vertices, scoring points. Painter then gives a color to a subset of that is independent in . The game ends when all vertices are colored. Painter and Lister want to minimize and maximize the total score, respectively. The best score that each player can guarantee is the sum-color cost of , written . The game is an online variant of online sum list coloring.We prove , where is the independence number, and we study when equality holds in the bounds. We compute for graphs with . Among -vertex trees, we prove that is minimized by the star and maximized by the path. We also study . 相似文献