共查询到20条相似文献,搜索用时 78 毫秒
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A graph with at least vertices is said to be distance -extendable if, for any matching of with edges in which the edges lie at distance at least pairwise, there exists a perfect matching of containing . In this paper we prove that every 5-connected triangulation on the projective plane of even order is distance 3 7-extendable and distance 4 -extendable for any . 相似文献
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We prove that a connected graph contains a circuit—a closed walk that repeats no edges—through any prescribed edges if and only if contains no odd cut of size at most . 相似文献
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Let be a planar graph with a list assignment . Suppose a preferred color is given for some of the vertices. We prove that if has girth at least six and all lists have size at least three, then there exists an -coloring respecting at least a constant fraction of the preferences. 相似文献
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Let be a finite -regular graph with a proper edge coloring. An edge Kempe switch is a new proper edge coloring of obtained by switching the two colors along some bichromatic cycle. We prove that any other edge coloring can be obtained by performing finitely many edge Kempe switches, provided that is replaced with a suitable finite covering graph. The required covering degree is bounded above by a constant depending only on . 相似文献
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Win conjectured that a graph on vertices contains disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least , where is even and . In this paper, we prove that Win's conjecture is true for , where is sufficiently large. To show this result, we prove a theorem on -factor in a graph under some Ore-type condition. Our main tools include Tutte's -factor theorem, the Karush-Kuhn-Tucker theorem on convex optimization and the solution to the long-standing 1-factor decomposition conjecture. 相似文献
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Louis DeBiasio Robert A. Krueger Dan Pritikin Eli Thompson 《Journal of Graph Theory》2020,94(1):92-112
Chen et al determined the minimum degree threshold for which a balanced -partite graph has a Hamiltonian cycle. We give an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary -partite graphs in that all parts have at most vertices (a necessary condition). To do this, we first prove a general result that both simplifies the process of checking whether a graph is a robust expander and gives useful structural information in the case when is not a robust expander. Then we use this result to prove that any -partite graph satisfying the minimum degree condition is either a robust expander or else contains a Hamiltonian cycle directly. 相似文献
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Hugo A. Akitaya Matthew D. Jones Matias Korman Oliver Korten Christopher Meierfrankenfeld Michael J. Munje Diane L. Souvaine Michael Thramann Csaba D. Tóth 《Journal of Graph Theory》2023,102(1):35-66
Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph and an integer , a -district map of is a partition of into nonempty subsets, called districts, each of which induces a connected subgraph of . A switch is an operation that modifies a -district map by reassigning a subset of vertices from one district to an adjacent district; a 1-switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all -district maps of a graph under 1-switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is PSPACE-complete to decide whether there exists a sequence of 1-switches that takes a given -district map into another; and NP-hard to find the shortest such sequence (even if a sequence of polynomial lengths is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that take a given -district map into another when the space is connected, and show that these algorithms perform a worst-case optimal number of switches up to constant factors. 相似文献
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The purpose of this note is to define a graph whose vertex set is a finite group , whose edge set is contained in that of the commuting graph of and contains the enhanced power graph of . We call this graph the deep commuting graph of . Two elements of are joined in the deep commuting graph if and only if their inverse images in every central extension of commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of act as automorphisms of the deep commuting graph. 相似文献
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The distinguishing index of a graph is the least cardinal number such that has an edge-coloring with colors, which is preserved only by the trivial automorphism. We prove a general upper bound for any connected infinite graph with finite maximum degree . This is in contrast with finite graphs since for every there exist infinitely many connected, finite graphs with . We also give examples showing that this bound is sharp for any maximum degree . 相似文献
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