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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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Yoshihiro Asayama Yuki Kawasaki Seog-Jin Kim Atsuhiro Nakamoto Kenta Ozeki 《Discrete Mathematics》2018,341(11):2988-2994
An -dynamic -coloring of a graph is a proper -coloring such that any vertex has at least distinct colors in . The -dynamic chromatic number of a graph is the least such that there exists an -dynamic -coloring of .Loeb et al. (2018) showed that if is a planar graph, then , and there is a planar graph with . Thus, finding an optimal upper bound on for a planar graph is a natural interesting problem. In this paper, we show that if is a planar triangulation. The upper bound is sharp. 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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The vertex arboricity of a graph is the minimum number of colors required to color the vertices of such that no cycle is monochromatic. The list vertex arboricity is the list-coloring version of this concept. In this note, we prove that if is a toroidal graph, then ; and if and only if contains as an induced subgraph. 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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Alexander Grigoriev Bert Marchal Natalya Usotskaya Ioan Todinca 《Discrete Applied Mathematics》2012,160(7-8):1262-1269
Given a graph with tree-width , branch-width , and side size of the largest square grid-minor , it is known that . In this paper, we introduce another approach to bound the side size of the largest square grid-minor specifically for planar graphs. The approach is based on measuring the distances between the faces in an embedding of a planar graph. We analyze the tightness of all derived bounds. In particular, we present a class of planar graphs where . 相似文献
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A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define for and We say that is -colorable if has a 2-coloring such that is an empty set or the induced subgraph has the maximum degree at most for and Let be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether is -colorable is NP-complete for every positive integer Moreover, we construct non--colorable planar graphs without 4-cycles and 5-cycles for every positive integer In contrast, we prove that is -colorable where and 相似文献
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Wei Xiong Jinquan Xu Zhengke Miao Yang Wu Hong-Jian Lai 《Discrete Mathematics》2017,340(12):2995-3001
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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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Let and be two positive integers such that and . A graph is an -parity factor of a graph if is a spanning subgraph of and for all vertices , and . In this paper we prove that every connected graph with vertices has an -parity factor if is even, , and for any two nonadjacent vertices , . This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998). 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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Haiyang Zhu Lianying Miao Sheng Chen Xinzhong Lü Wenyao Song 《Discrete Mathematics》2018,341(8):2211-2219
Let be the set of all positive integers. A list assignment of a graph is a function that assigns each vertex a list for all . We say that is --choosable if there exists a function such that for all , if and are adjacent, and if and are at distance 2. The list--labeling number of is the minimum such that for every list assignment , is --choosable. We prove that if is a planar graph with girth
and its maximum degree is large enough, then . There are graphs with large enough and having . 相似文献
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A graph is minimally -tough if the toughness of is and the deletion of any edge from decreases the toughness. Kriesell conjectured that for every minimally -tough graph the minimum degree . We show that in every minimally -tough graph . We also prove that every minimally -tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number any graph can be embedded as an induced subgraph into a minimally -tough graph. 相似文献
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Jakub Przybyło 《Discrete Mathematics》2018,341(4):1098-1102
Let be a (not necessarily proper) total colouring of a graph with maximum degree . Two vertices are sum distinguished if they differ with respect to sums of their incident colours, i.e. . The least integer admitting such colouring under which every at distance in
are sum distinguished is denoted by . Such graph invariants link the concept of the total vertex irregularity strength of graphs with so-called 1-2-Conjecture, whose concern is the case of . Within this paper we combine probabilistic approach with purely combinatorial one in order to prove that for every integer and each graph , thus improving the previously best result: . 相似文献