共查询到20条相似文献,搜索用时 593 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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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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Let be a finite group, written multiplicatively. The Davenport constant of is the smallest positive integer such that every sequence of with elements has a non-empty subsequence with product . Let be the Dihedral Group of order and be the Dicyclic Group of order . Zhuang and Gao (2005) showed that and Bass (2007) showed that . In this paper, we give explicit characterizations of all sequences of such that and is free of subsequences whose product is 1, where is equal to or for some . 相似文献
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In this paper, we show that for any fixed integers and , the star-critical Ramsey number for all sufficiently large . Furthermore, for any fixed integers and , as . 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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Let be a -connected graph of order . In [1], Bondy (1980) considered a degree sum condition for a graph to have a Hamiltonian cycle, say, to be covered by one cycle. He proved that if , then has a Hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. (1995) [4] proved that if and , then can be covered by two cycles. By these results, we conjecture that if , then can be covered by two cycles. In this paper, we prove the case of this conjecture. In fact, we prove a stronger result; if is 2-connected with , then can be covered by two cycles, or belongs to an exceptional class. 相似文献
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In this article, we prove that the compact simple Lie groups for , for , for , , and admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov. 相似文献
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Let and denote the maximum degree and the Laplacian spectral radius of a tree , respectively. In this paper we prove that for two trees and on vertices, if and , then , and the bound “” is the best possible. We also prove that for two trees and on vertices with perfect matchings, if and , then . 相似文献
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Vahan V. Mkrtchyan Samvel S. Petrosyan Gagik N. Vardanyan 《Discrete Mathematics》2010,310(10-11):1588-1613
For and a cubic graph let denote the maximum number of edges that can be covered by matchings. We show that and . Moreover, it turns out that . 相似文献
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Let be a set of at least two vertices in a graph . A subtree of is a -Steiner tree if . Two -Steiner trees and are edge-disjoint (resp. internally vertex-disjoint) if (resp. and ). Let (resp. ) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) -Steiner trees in , and let (resp. ) be the minimum (resp. ) for ranges over all -subset of . Kriesell conjectured that if for any , then . He proved that the conjecture holds for . In this paper, we give a short proof of Kriesell’s Conjecture for , and also show that (resp. ) if (resp. ) in , where . Moreover, we also study the relation between and , where is the line graph of . 相似文献