共查询到20条相似文献,搜索用时 31 毫秒
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Susan A. van Aardt Christoph Brause Alewyn P. Burger Marietjie Frick Arnfried Kemnitz Ingo Schiermeyer 《Discrete Mathematics》2017,340(11):2673-2677
An edge-coloured graph is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph denoted by , is the smallest number of colours that are needed in order to make properly connected. Our main result is the following: Let be a connected graph of order and . If , then except when and where and 相似文献
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On stable cutsets in claw-free graphs and planar graphs 总被引:4,自引:0,他引:4
A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let and (claw) denote the complete (bipartite) graph on 4 and vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a -free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free graph with maximum degree at most four or whether a (claw, )-free graph admits a stable cutset. As a by-product we obtain that the stable cutset problem is polynomially solvable for claw-free planar graphs, and also for planar line graphs.Thus, the computational complexity of the stable cutset problem is completely determined for claw-free graphs with respect to degree constraint, and for claw-free planar graphs. Moreover, we prove that the stable cutset problem remains NP-complete for -free planar graphs with maximum degree five. 相似文献
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This paper considers a degree sum condition sufficient to imply the existence of vertex-disjoint cycles in a graph . For an integer , let be the smallest sum of degrees of independent vertices of . We prove that if has order at least and , with , then contains vertex-disjoint cycles. We also show that the degree sum condition on is sharp and conjecture a degree sum condition on sufficient to imply contains vertex-disjoint cycles for . 相似文献
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A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs
Let be a balanced bipartite graph of order , and let be the minimum degree sum of two non-adjacent vertices in different partite sets of . In 1963, Moon and Moser proved that if , then is hamiltonian. In this note, we show that if is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly cycles for sufficiently large graphs. In order to prove this, we also give a condition for the existence of vertex-disjoint alternating cycles with respect to a chosen perfect matching in . 相似文献
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Let be a graph of order . An even squared Hamiltonian cycle (ESHC) of is a Hamiltonian cycle of with chords for all (where for ). When is even, an ESHC contains all bipartite -regular graphs of order . We prove that there is a positive integer such that for every graph of even order , if the minimum degree is , then contains an ESHC. We show that the condition of being even cannot be dropped and the constant cannot be replaced by . Our results can be easily extended to even th powered Hamiltonian cycles for all . 相似文献
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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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Let denote the chromatic polynomial of a graph on vertices. The ‘shameful conjecture’ due to Bartels and Welsh states that, Let denote the expected number of colors used in a uniformly random proper -coloring of . The above inequality can be interpreted as saying that , where is the empty graph on nodes. This conjecture was proved by F.M. Dong, who in fact showed that, for all . There are examples showing that this inequality is not true for all . In this paper, we show that the above inequality holds for all , where is the largest degree of . It is also shown that the above inequality holds true for all when is a claw-free graph. 相似文献
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For bipartite graphs , the bipartite Ramsey number is the least positive integer so that any coloring of the edges of with colors will result in a copy of in the th color for some . In this paper, our main focus will be to bound the following numbers: and for all for and for Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result. 相似文献
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