共查询到20条相似文献,搜索用时 531 毫秒
1.
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 -coloring of a graph with colors is a proper coloring of using colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer for which has a -coloring using colors is the -chromatic number of . The -spectrum of a graph is the set of positive integers , for which has a -coloring using colors. A graph is -continuous if = the closed interval . In this paper, we obtain an upper bound for the -chromatic number of some families of Kneser graphs. In addition we establish that for the Kneser graph whenever . We also establish the -continuity of some families of regular graphs which include the family of odd graphs. 相似文献
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Let be a connected regular graph and , , the line, subdivision, total graphs of , respectively. In this paper, we derive formulae and lower bounds of the Kirchhoff index of , and , respectively. In particular, we give special formulae for the Kirchhoff index of , and , where is a complete graph , a regular complete bipartite graph and a cycle . 相似文献
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Li-Da Tong 《Discrete Mathematics》2009,309(12):4205-4207
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We say a graph is -colorable with of ’s and of ’s if may be partitioned into independent sets and sets whose induced graphs have maximum degree at most . The maximum average degree, , of a graph is the maximum average degree over all subgraphs of . In this note, for nonnegative integers , we show that if , then is -colorable. 相似文献
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For a graph let , and denote its independence number, matching number, and vertex cover number, respectively. If or, equivalently, , then is a König–Egerváry graph.In this paper we give a new characterization of König–Egerváry graphs. 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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Let be a graph. A set is a restrained dominating set if every vertex not in is adjacent to a vertex in and to a vertex in . The restrained domination number of , denoted by , is the smallest cardinality of a restrained dominating set of . We define the restrained bondage number of a nonempty graph to be the minimum cardinality among all sets of edges for which . Sharp bounds are obtained for , and exact values are determined for several classes of graphs. Also, we show that the decision problem for is NP-complete even for bipartite graphs. 相似文献
11.
Andrzej Grzesik 《Discrete Mathematics》2012,312(23):3467-3472
We study a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent realization of this project. The smallest number of colors necessary for Ann to win the game on a graph (regardless of Ben’s strategy) is called the indicated chromatic number of , and denoted by . We approach the question how much differs from the usual chromatic number . In particular, whether there is a function such that for every graph . We prove that cannot be linear with leading coefficient less than . On the other hand, we show that the indicated chromatic number of random graphs is bounded roughly by . We also exhibit several classes of graphs for which and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs. 相似文献
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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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Ronald J. Gould Wenliang Tang Erling Wei Cun-Quan Zhang 《Discrete Mathematics》2012,312(17):2682-2689
Let be a simple graph. A graph is called an -saturated graph if is not a subgraph of , but adding any missing edge to will produce a copy of . Denote by the set of all -saturated graphs with order . Then the saturation number is defined as , and the extremal number is defined as . A natural question is that of whether we can find an -saturated graph with edges for any . The set of all possible values is called the edge spectrum for -saturated graphs. In this paper we investigate the edge spectrum for -saturated graphs, where . It is trivial for the case of that the saturated graph must be an empty graph. 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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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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A graph has a perfect matching if and only if is not a root of its matching polynomial . Thus, Tutte’s famous theorem asserts that is not a root of if and only if for all , where denotes the number of odd components of . Tutte’s theorem can be proved using a characterization of the structure of maximal non-matchable graphs, that is, the edge-maximal graphs among those having no perfect matching. In this paper, we prove a generalized version of Tutte’s theorem in terms of avoiding any given real number as a root of . We also extend maximal non-matchable graphs to maximal -non-matchable graphs and determine the structure of such graphs. 相似文献
18.
Yehong Shao 《Discrete Mathematics》2018,341(12):3441-3446
Let be a graph and be its line graph. In 1969, Chartrand and Stewart proved that , where and denote the edge connectivity of and respectively. We show a similar relationship holds for the essential edge connectivity of and , written and , respectively. In this note, it is proved that if is not a complete graph and does not have a vertex of degree two, then . An immediate corollary is that for such graphs , where the vertex connectivity of the line graph
and the second iterated line graph are written as and respectively. 相似文献
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Hongliang Lu 《Discrete Applied Mathematics》2013,161(13-14):2075-2078
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Let be a finite simple graph. For , the difference of , where is the neighborhood of and is called the critical difference of . is called a critical set if equals the critical difference and is the intersection of all critical sets. is the union of all critical independent sets. An independent set is an inclusion minimal set with if no proper subset of has positive difference.A graph is called a König–Egerváry graph if the sum of its independence number and matching number equals .In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set with is at least the critical difference of the graph.We also give a new short proof of the inequality .A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph , the critical difference equals , is proved.We also make an observation about using Edmonds–Gallai Structure Theorem as a concluding remark. 相似文献