共查询到20条相似文献,搜索用时 78 毫秒
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In this paper, we study a new coloring parameter of graphs called the gap vertex-distinguishing edge coloring. It consists in an edge-coloring of a graph which induces a vertex distinguishing labeling of such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distinguishing edge coloring of is called the gap chromatic number of and is denoted by .We here study the gap chromatic number for a large set of graphs of order and prove that . 相似文献
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《Discrete Mathematics》2018,341(10):2878-2882
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Carl Johan Casselgren Hrant H. Khachatrian Petros A. Petrosyan 《Discrete Mathematics》2018,341(3):627-637
An interval-coloring of a multigraph is a proper edge coloring with colors such that the colors of the edges incident with every vertex of are colored by consecutive colors. A cyclic interval-coloring of a multigraph is a proper edge coloring with colors such that the colors of the edges incident with every vertex of are colored by consecutive colors, under the condition that color is considered as consecutive to color . Denote by () and () the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph , respectively. We present some new sharp bounds on and for multigraphs satisfying various conditions. In particular, we show that if is a -connected multigraph with an interval coloring, then . We also give several results towards the general conjecture that for any triangle-free graph with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most . 相似文献
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Bojan Vučković 《Discrete Mathematics》2018,341(3):820-824
Let be a graph without isolated edges, and let be a coloring of the edges, where adjacent edges may be colored the same. The color code of a vertex is the ordered -tuple , where is the number of edges incident with that are colored . If every two adjacent vertices of have different color codes, such a coloring is called multi-set neighbor distinguishing. In this paper, we prove that three colors are sufficient to produce a multi-set neighbor distinguishing edge coloring for every graph without isolated edges. 相似文献
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Amel Bendali-Braham Noureddine Ikhlef-Eschouf Mostafa Blidia 《Comptes Rendus Mathematique》2018,356(2):115-120
The b-chromatic number of a graph G is the largest integer k such that G admits a proper coloring with k colors for which each color class contains a vertex that has at least one neighbor in all the other color classes. A graph G is called -critical if the contraction of any edge e of G decreases the b-chromatic number of G. The purpose of this paper is the characterization of all -critical trees. 相似文献
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Let be a connected graph. A configuration of pebbles on is a function that assigns a nonnegative integer to each vertex. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A configuration is solvable if after making pebbling moves any vertex can get at least one pebble. The pebbling number of , denoted , is the smallest integer such that any configuration of pebbles on is solvable. A graph has the two-pebbling property if after placing more than pebbles on , where is the number of vertices with pebbles, there is a sequence of pebbling moves so that at least two pebbles can be placed on any vertex. A graph without the two-pebbling property is called a Lemke graph. Previously, an infinite family of Lemke graphs was shown to exist by subdividing edges of the original Lemke graph. In this paper, we introduce a new way to create infinite families of Lemke graphs based on adding vertices as well as subdividing edges. We also characterize the configurations that violate the two-pebbling property on these graphs and conjecture another infinite family of Lemke graphs that generalizes the original Lemke graph. 相似文献
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Given a graph , a set is a dominating set of if every vertex of is either in or adjacent to a vertex in . The domination number of , denoted , is the minimum cardinality of a dominating set of . Vizing’s conjecture states that for any graphs and where denotes the Cartesian product of and . In this paper, we continue the work by Anderson et al. (2016) by studying the domination number of the hierarchical product. Specifically, we show that partitioning the vertex set of a graph in a particular way shows a trend in the lower bound of the domination number of the product, providing further evidence that the conjecture is true. 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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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 power graph of a finite group is the graph whose vertex set is , two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of and characterize the groups achieving the bounds. Moreover, we determine the independence number of if is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups whose power graphs have independence number 3 or , where is the order of . 相似文献
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Christina J. Edholm Leslie Hogben My Huynh Joshua LaGrange Darren D. Row 《Linear algebra and its applications》2012,436(12):4352-4372
The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for ) is nonzero whenever is an edge in G and is zero otherwise; maximum nullity is taken over the same set of matrices. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. The spread of a graph parameter at a vertex v or edge e of G is the difference between the value of the parameter on G and on or . Rank spread (at a vertex) was introduced in [4]. This paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank/maximum nullity and zero forcing number. Properties of the spreads are established and used to determine values of the minimum rank/maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted. 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 transition in a graph is defined as a pair of adjacent edges. A transition system of an Eulerian graph refers to a set of partitions such that for each vertex of the graph, there corresponds to a partition of the set of edges incident to the vertex into transitions. A generalized transition system over a graph defines a set of transitions over . A compatible Eulerian circuit of an Eulerian graph with a generalized transition system is defined as an Eulerian circuit in which no two consecutive edges form a transition defined by . In this paper, we further introduce the concept of weakly generalized transition system which is an extension of the generalized transition system and prove some Ore-type sufficient conditions for the existence of compatible Eulerian circuits in Eulerian graphs with (weakly) generalized transition systems and obtain corresponding results for Eulerian digraphs. Our conditions improve some previous results due to Jackson and Isaak, respectively. 相似文献
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David S. Herscovici Benjamin D. Hester Glenn H. Hurlbert 《Discrete Mathematics》2012,312(15):2286-2293
We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that , where is the pebbling number of the graph . We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that shows that Sjöstrand’s theorem on cover pebbling does not apply if we allow the cost of transferring a pebble from one vertex to an adjacent vertex to depend on the weight of the edge and we describe an alternate pebbling number for which Graham’s conjecture is demonstrably false. 相似文献
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The inflation of a graph is obtained from by replacing every vertex of degree by a clique and each edge by an edge between two vertices of the corresponding cliques and of in such a way that the edges of which come from the edges of form a matching of . A set of vertices in a graph is a total dominating set, abbreviated TDS, of if every vertex of is adjacent to a vertex in . The minimum cardinality of a TDS of is the total domination number of . In this paper, we investigate total domination in inflated graphs. We provide an upper bound on the total domination number of an inflated graph in terms of its order and matching number. We show that if is a connected graph of order , then , and we characterize the graphs achieving equality in this bound. Further, if we restrict the minimum degree of to be at least , then we show that , with equality if and only if has a perfect matching. If we increase the minimum degree requirement of to be at least , then we show , with equality if and only if every minimum TDS of is a perfect total dominating set of , where a perfect total dominating set is a TDS with the property that every vertex is adjacent to precisely one vertex of the set. 相似文献