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1.
An antimagic labeling of a digraph with vertices and arcs is a bijection from the set of arcs of to such that all oriented vertex sums are pairwise distinct, where an oriented vertex sum of a vertex is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. Hefetz, Mütze and Schwartz conjectured every connected undirected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that every Halin graph admits an antimagic orientation. 相似文献
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Let be an integer and be a graph. Let denote the graph obtained from by replacing each edge of with parallel edges. We say that has all -factors or all fractional -factors if has an -factor or a fractional -factor for every function with even. In this note, we come up with simple characterizations of a graph such that has all -factors or all fractional -factors. These characterizations are extensions of Tutte’s 1-Factor Theorem and Tutte’s Fractional 1-Factor Theorem. 相似文献
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《Discrete Mathematics》2019,342(2):339-343
A strong edge-coloring of a graph is a partition of its edge set into induced matchings. Let be a connected planar graph with girth and maximum degree . We show that either is isomorphic to a subgraph of a very special -regular graph with girth , or has a strong edge-coloring using at most colors. 相似文献
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Kiyoshi Ando 《Discrete Mathematics》2019,342(12):111598
An edge of a -connected graph is said to be -contractible if the contraction of the edge results in a -connected graph. For a graph and a vertex of , let be the subgraph induced by the neighborhood of . We prove that if has less than edges for any vertex of a -connected graph , then has a -contractible edge. We also show that the bound is sharp. 相似文献
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In 2009, Kyaw proved that every -vertex connected -free graph with contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw’s result for connected -free graphs. We show that every -vertex connected -free graph with contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition “” is best possible. 相似文献
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A of a digraph with arcs is a bijection from the set of arcs of to . A labeling of is if no two vertices in have the same vertex-sum, where the vertex-sum of a vertex for a labeling is the sum of labels of all arcs entering minus the sum of labels of all arcs leaving . An orientation of a graph is if has an antimagic labeling. Hefetz et al. (2010) raised the question: Does every graph admit an antimagic orientation? It had been proved that every 2-regular graph with at most two odd components has an antimagic orientation. In this paper, we consider 2-regular graphs with more than two odd components. We show that every 2-regular graph with odd components has an antimagic orientation. And we show that each 2-regular graph with odd components admits an antimagic orientation if each odd component has at least vertices with . 相似文献
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《Discrete Mathematics》2020,343(12):112117
Let be an edge-colored graph of order . The minimum color degree of , denoted by , is the largest integer such that for every vertex , there are at least distinct colors on edges incident to . We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if , then contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if and , then contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if , then contains vertex-disjoint rainbow triangles. For any integer , we show that if and , then contains vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of edge-disjoint rainbow triangles. 相似文献
9.
Christian Bosse 《Discrete Mathematics》2019,342(12):111595
The Hadwiger number of a graph , denoted , is the largest integer such that contains as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph , , where denotes the chromatic number of . Let denote the independence number of . A graph is -free if it does not contain the graph as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that for all -free graphs with , where is any graph on four vertices with , , or is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs on five vertices with . In this note, we prove that for all -free graphs with , where denotes the wheel on six vertices. 相似文献
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For an integer , a graph is -hamiltonian if for any vertex subset with , is hamiltonian, and is -hamiltonian connected if for any vertex subset with , is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Ku?zel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjá?ek and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of -hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for , a line graph is -hamiltonian if and only if is -connected. In this paper we prove the following.(i) For an integer , the line graph of a claw-free graph is -hamiltonian if and only if is -connected.(ii) The line graph of a claw-free graph is 1-hamiltonian connected if and only if is 4-connected. 相似文献
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《Discrete Mathematics》2019,342(3):777-783
Let be a connected graph. A configuration of pebbles assigns a nonnegative integer number of pebbles to each vertex of . A move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A configuration is solvable if any vertex can get at least one pebble through a sequence of moves. 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 moves so that at least two pebbles can be placed on any vertex. A graph has the odd-two-pebbling property if after placing more than pebbles on , where is the number of vertices with an odd number of pebbles, there is a sequence of moves so that at least two pebbles can be placed on any vertex. In this paper, we prove that the two-pebbling and odd-two-pebbling properties are not equivalent. 相似文献
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《Discrete Mathematics》2022,345(4):112784
A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number of G is the minimum cardinality of a dominating set in G, while the total domination number of G is the minimum cardinality of total dominating set in G. A claw-free graph is a graph that does not contain as an induced subgraph. Let G be a connected, claw-free, cubic graph of order n. We show that if we exclude two graphs, then , and this bound is best possible. In order to prove this result, we prove that if we exclude four graphs, then , and this bound is best possible. These bounds improve previously best known results due to Favaron and Henning (2008) [7], Southey and Henning (2010) [19]. 相似文献
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《Discrete Mathematics》2022,345(8):112903
Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number is the least integer k for which G admits a coloring with k colors such that each color class induces a -degenerate subgraph of G. So is the chromatic number and is the point arboricity. The point partition number with was introduced by Lick and White. A graph G is called -critical if every proper subgraph H of G satisfies . In this paper we prove that if G is a -critical graph whose order satisfies , then G can be obtained from two non-empty disjoint subgraphs and by adding t edges between any pair of vertices with and . Based on this result we establish the minimum number of edges possible in a -critical graph G of order n and with , provided that and t is even. For the corresponding two results were obtained in 1963 by Tibor Gallai. 相似文献
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A graph is -colorable if it admits a vertex partition into a graph with maximum degree at most and a graph with maximum degree at most . We show that every -free planar graph is -colorable. We also show that deciding whether a -free planar graph is -colorable is NP-complete. 相似文献
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The tensor product of graphs , and is defined by and Let be the fractional chromatic number of a graph . In this paper, we prove that if one of the three graphs , and is a circular clique, 相似文献
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Let be a simple connected graph with vertices and edges. The spectral radius of is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of which improves some known bounds: If , where is an integer, then The equality holds if and only if is a complete graph or , where is the graph obtained from by deleting some edge . 相似文献
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Jakub Przybyło 《Discrete Mathematics》2019,342(2):498-504
Let be any graph without isolated edges. The well known 1–2–3 Conjecture asserts that the edges of can be weighted with so that adjacent vertices have distinct weighted degrees, i.e. the sums of their incident weights. It was independently conjectured that if additionally has no isolated triangles, then it can be edge decomposed into two subgraphs which fulfil the 1–2–3 Conjecture with just weights 1,2, i.e. such that there exist weightings so that for every , if then , where denotes the sum of weights incident with in for . We apply the probabilistic method to prove that the known weakening of this so-called Standard (2,2)-Conjecture holds for graphs with minimum degree large enough. Namely, we prove that if , then can be decomposed into graphs for which weightings exist so that for every , or . In fact we prove a stronger result, as one of the weightings is redundant, i.e. uses just weight 1. 相似文献
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For given graphs , , the -color Ramsey number, denoted by , is the smallest integer such that if we arbitrarily color the edges of a complete graph of order with colors, then it always contains a monochromatic copy of colored with , for some . Let be a cycle of length and a star of order . In this paper, firstly we give a general upper bound of . In particular, for the 3-color case, we have and this bound is tight in some sense. Furthermore, we prove that for all and , and if is a prime power, then the equality holds. 相似文献
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