共查询到20条相似文献,搜索用时 15 毫秒
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For a positive integer , a graph is -knitted if for each subset of vertices, and every partition of into (disjoint) parts for some , one can find disjoint connected subgraphs such that contains for each . In this article, we show that if the minimum degree of an -vertex graph is at least when , then is -knitted. The minimum degree is sharp. As a corollary, we obtain that -contraction-critical graphs are -connected. 相似文献
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Nathan Linial 《Discrete Mathematics》1976,15(3):297-300
Let G=(V, E) be a block of order n, different from Kn. Let m=min {d(x)+d(y): [x, y]?E}. We show that if m?n then G contains a cycle of length at least m. 相似文献
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Let G be a connected graph of order p ≥ 2, with edge-connectivity κ1(G) and minimum degree δ(G). It is shown her ethat in order to obtain the equality κ1(G) = δ(G), it is sufficient that, for each vertex x of minimum degree in G, the vertices in the neighbourhood N(x) of x have sufficiently large degree sum. This result implies a previous result of Chartrand, which required that δ(G) ≥ [p/2]. 相似文献
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Matt Devos Zdeněk Dvořák Jacob Fox Jessica McDonald Bojan Mohar Diego Scheide 《Combinatorica》2014,34(3):279-298
An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P uv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2, where c>0 is an absolute constant. For small values of t, 1≤t≤7, every simple graph of minimum degree at least t?1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large. 相似文献
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Let G be a graph of order n, and let a and b be integers such that 1 ≤ a < b. We show that G has an [a, b]-factor if δ(G) ≥ a, n ≥ 2a + b + and max {dG(u), dG(v) ≥ for any two nonadjacent vertices u and v in G. This result is best possible, and it is an extension of T. Iida and T. Nishimura's results (T. Iida and T. Nishimura, An Ore-type condition for the existence of k-factors in graphs, Graphs and Combinat. 7 (1991), 353–361; T. Nishimura, A degree condition for the existence of k-factors, J. Graph Theory 16 (1992), 141–151). about the existence of a k-factor. As an immediate consequence, it shows that a conjecture of M. Kano (M. Kano, Some current results and problems on factors of graphs, Proc. 3rd China–USA International Conference on Graph Theory and Its Application, Beijing (1993). about connected [a, b]-factors is incorrect. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 1–6, 1998 相似文献
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李红燕 《纯粹数学与应用数学》2016,32(2):127-131
连通图G的一个k-树是指图G的一个最大度至多是k的生成树.对于连通图G来说,其毁裂度定义为r(G)=max{ω(G-X)-|X|-m(G-X)|X■V(G),ω(G-X)1}其中ω(G-X)和m(G-X)分别表示G-X中的分支数目和最大分支的阶数.本文结合毁裂度给出连通图G包含一个k-树的充分条件;利用图的结构性质和毁裂度的关系逐步刻画并给出图G包含一个k-树的毁裂度条件. 相似文献
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F.T Boesch 《Journal of Combinatorial Theory, Series B》1974,16(2):162-165
It is shown that Bondy's degree condition for n-connectedness of a graph is the strongest monotone degree condition for forcibly n-connected graphic sequences. 相似文献
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The celebrated result of Fleischner states that the square of every 2-connected graph is Hamiltonian. We investigate what happens if the graph is just connected. For every n ≥ 3, we determine the smallest length c(n) of a longest cycle in the square of a connected graph of order n and show that c(n) is a logarithmic function in n. Furthermore, for every c ≥ 3, we characterize the connected graphs of largest order whose square contains no cycle of length at least c. 相似文献
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Komlós (2000) determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree. 相似文献
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Dr. Norbert Köhler 《Monatshefte für Mathematik》1981,92(2):105-116
Those non-hamiltonian graphsG withn vertices are characterized, which satisfy the Ore-type degree-conditiond(x)+d(y)n–2 for each pairx,yM of different nonadjacent vertices whereM consists of two vertices ofG. As an application a theorem on hamiltonian connectivity of graphs is given. Furthermore, a condition is presented which is sufficient for the existence of a covering of a graph by two disjoint paths with prescribed set of startpoints and prescribed set of endpoints. A class of graphs is described which have no covering of this kind. 相似文献
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R.J. Faudree 《Discrete Mathematics》2009,309(19):5891-1061
Let G be a graph of order n and circumference c(G). Let be the complement of G. We prove that and show sharpness of this bound. 相似文献
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Tsuyoshi Nishimura 《Journal of Graph Theory》1992,16(2):141-151
Let k be an integer such that ≦, and let G be a connected graph of order n with ≦, kn even, and minimum degree at least k. We prove that if G satisfies max(deg(u), deg(v)) ≦ n/2 for each pair of nonadjacent vertices u, v in G, then G has a k-factor. 相似文献
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P. Katerinis obtained a sufficient condition for the existence of a 2-factor in a bipartite graph, in the spirit of Hall's theorem. We show a sufficient condition for the existence of a k-factor in a bipartite graph, as a generalization of Katerinis's theorem and Hall's theorem. 相似文献
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If G is a connected graph with vertex set V, then the degree distance of G, D′(G), is defined as , where degw is the degree of vertex w, and d(u,v) denotes the distance between u and v. We prove the asymptotically sharp upper bound for graphs of order n and diameter d. As a corollary we obtain the bound for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163]. 相似文献
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The core GΔ of a simple graph G is the subgraph induced by the vertices of maximum degree. It is well known that the Petersen graph is not 1-factorizable and has property that the core of the graph obtained from it by removing one vertex has maximum degree 2. In this paper, we prove the following result. Let G be a regular graph of even order with d(G) ≥ 3. Suppose that G contains a vertex ν such that the core of G\ν has maximum degree 2. If G is not the Petersen graph, then G is 1-factorizable. © 1993 John Wiley & Sons, Inc. 相似文献