共查询到20条相似文献,搜索用时 31 毫秒
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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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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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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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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