共查询到20条相似文献,搜索用时 687 毫秒
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The vertex arboricity of a graph is the minimum number of colors the vertices can be labeled so that each color class induces a forest. It was well-known that for every planar graph . In this paper, we prove that if is a planar graph without 7-cycles. This extends a result in [A. Raspaud, W. Wang, On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008) 1064–1075] that for each , planar graphs without -cycles have . 相似文献
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An -dynamic -coloring of a graph is a proper -coloring such that for any vertex , there are at least distinct colors in . The -dynamic chromatic number of a graph is the least such that there exists an -dynamic -coloring of . The list-dynamic chromatic number of a graph is denoted by .Recently, Loeb et al. (0000) showed that the list -dynamic chromatic number of a planar graph is at most 10. And Cheng et al. (0000) studied the maximum average condition to have , or . On the other hand, Song et al. (2016) showed that if is planar with girth at least 6, then for any .In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that if , if , and if . All of the bounds are tight. 相似文献
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Kathie Cameron Murilo V.G. da Silva Shenwei Huang Kristina Vušković 《Discrete Mathematics》2018,341(2):463-473
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph has a vertex of degree at most , and hence , where denotes the size of a largest clique in and denotes the chromatic number of . We give an algorithm for -coloring these graphs for fixed and an algorithm for maximum weight stable set, where is the number of vertices and is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring.Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs without clique cutsets have treewidth at most and clique-width at most 48. 相似文献
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Bojan Vučković 《Discrete Mathematics》2018,341(5):1472-1478
An adjacent vertex distinguishing total -coloring of a graph is a proper total -coloring of such that any pair of adjacent vertices have different sets of colors. The minimum number needed for such a total coloring of is denoted by . In this paper we prove that if , and in general. This improves a result in Huang et al. (2012) which states that for any graph with . 相似文献
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The neighbor-distinguishing total chromatic number of a graph is the smallest integer such that can be totally colored using colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph with maximum degree 13 to have or . Precisely, we show that if is a planar graph of maximum degree 13, then ; and if and only if contains two adjacent 13-vertices. 相似文献
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For a subgraph of , let be the maximum number of vertices of that are pairwise distance at least three in . In this paper, we prove three theorems. Let be a positive integer, and let be a subgraph of an -connected claw-free graph . We prove that if , then either can be covered by a cycle in , or there exists a cycle in such that . This result generalizes the result of Broersma and Lu that has a cycle covering all the vertices of if . We also prove that if , then either can be covered by a path in , or there exists a path in such that . By using the second result, we prove the third result. For a tree , a vertex of with degree one is called a leaf of . For an integer , a tree which has at most leaves is called a -ended tree. We prove that if , then has a -ended tree covering all the vertices of . This result gives a positive answer to the conjecture proposed by Kano et al. (2012). 相似文献
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Vahan V. Mkrtchyan Samvel S. Petrosyan Gagik N. Vardanyan 《Discrete Mathematics》2010,310(10-11):1588-1613
For and a cubic graph let denote the maximum number of edges that can be covered by matchings. We show that and . Moreover, it turns out that . 相似文献
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A star edge-coloring of a graph is a proper edge coloring such that every 2-colored connected subgraph of is a path of length at most 3. For a graph , let the list star chromatic index of , , be the minimum such that for any -uniform list assignment for the set of edges, has a star edge-coloring from . Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph is less than (resp. 3), then (resp. ). 相似文献
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Bojan Vučković 《Discrete Mathematics》2017,340(12):3092-3096
A proper edge coloring is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The minimum number of colors needed for a neighbor-distinguishing edge coloring is the neighbor-distinguishing index, denoted by . A graph is normal if it contains no isolated edges. Let be a normal graph, and let and denote the maximum degree and the chromatic index of , respectively. We modify the previously known techniques of edge-partitioning to prove that , which implies that . This improves the result in Wang et al. (2015), which states that for any normal graph. We also prove that when , is an integer with . 相似文献
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