共查询到20条相似文献,搜索用时 15 毫秒
1.
《Discrete Mathematics》2019,342(6):1546-1552
2.
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
3.
In this paper, we introduce and study a generalization of the degree constrained minimum spanning tree problem where we may install one of several available transmission systems (each with a different cost value) in each edge. The degree of the endnodes of each edge depends on the system installed on the edge. We also discuss a particular case that arises in the design of wireless mesh networks (in this variant the degree of the endnodes of each edge depend on the transmission system installed on it as well as on the length of the edge). We propose three classes of models using different sets of variables and compare from a theoretical perspective as well as from a computational point of view, the models and the corresponding linear programming relaxations. The computational results show that some of the proposed models are able to solve to optimality instances with 100 nodes and different scenarios. 相似文献
4.
《Discrete Mathematics》2022,345(9):112966
A broom is a tree obtained by identifying an endpoint of a path with the center of a star. Let G be a connected graph of order . Chen et al. [2] conjectured that if the degree sum is at least for any three pairwise nonadjacent vertices, then G contains a spanning broom. In this paper, we confirm the conjecture for . 相似文献
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David Zuckerman 《Journal of Theoretical Probability》1989,2(1):147-157
The motivating problem for this paper is to find the expected covering time of a random walk on a balanced binary tree withn vertices. Previous upper bounds for general graphs ofO(|V| |E|)(1) andO(|V| |E|/d
min)(2) imply an upper bound ofO(n
2). We show an upper bound on general graphs ofO( |E| log |V|), which implies an upper bound ofO(n log2
n). The previous lower bound was (|V| log |V|) for trees.(2) In our main result, we show a lower bound of (|V| (log
d
max |V|)2) for trees, which yields a lower bound of (n log2
n). We also extend our techniques to show an upper bound for general graphs ofO(max{E
Ti} log |V|). 相似文献
7.
Tomoki Yamashita 《Discrete Mathematics》2008,308(9):1620-1627
For a graph G, let σk(G) be the minimum degree sum of an independent set of k vertices. Ore showed that if G is a graph of order n?3 with σ2(G)?n then G is hamiltonian. Let κ(G) be the connectivity of a graph G. Bauer, Broersma, Li and Veldman proved that if G is a 2-connected graph on n vertices with σ3(G)?n+κ(G), then G is hamiltonian. On the other hand, Bondy showed that if G is a 2-connected graph on n vertices with σ3(G)?n+2, then each longest cycle of G is a dominating cycle. In this paper, we prove that if G is a 3-connected graph on n vertices with σ4(G)?n+κ(G)+3, then G contains a longest cycle which is a dominating cycle. 相似文献
8.
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
9.
Tomoki Yamashita 《Discrete Mathematics》2008,308(24):6584-6587
Let G be a graph. For S⊂V(G), let Δk(S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If Δ2(S)≥d for every independent set S of order κ(G)+1, then G has a cycle of length at least min{d,|V(G)|}. (2) Let G be a 2-connected graph and X a subset of V(G). If Δ2(S)≥|V(G)| for every independent set S of order κ(X)+1 in G[X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles. 相似文献
10.
A graph is called supereulerian if it has a spanning closed trail. Let G be a 2-edge-connected graph of order n such that each minimal edge cut SE(G) with |S|3 satisfies the property that each component of G−S has order at least (n−2)/5. We prove that either G is supereulerian or G belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ4: If G is a 2-edge-connected graph of order n with δ(G)4 such that for every edge xyE(G) , we have max{d(x),d(y)}(n−2)/5−1, then either G is supereulerian or G belongs to one of two classes of exceptional graphs. We show that the condition δ(G)4 cannot be relaxed. 相似文献
11.
Shengning Qiao 《Discrete Mathematics》2009,309(13):4642-4645
Let F be an oriented forest with n vertices and m arcs and D be a digraph without loops and multiple arcs. In this note we prove that D contains a subdigraph isomorphic to F if D has at least n vertices and min{d+(u)+d+(v),d−(u)+d−(v),d+(u)+d−(v)}≥2m−1 for every pair of vertices u,v∈V(D) with uv∉A(D). This is a common generalization of two results of Babu and Diwan, one on the existence of forests in graphs under a degree sum condition and the other on the existence of oriented forests in digraphs under a minimum degree condition. 相似文献
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Genzhen Yu 《Discrete Mathematics》2011,(1):38
The Aztec diamond was extensively studied in both graph theory and statistical physics. Knuth obtained a formula of the number of spanning trees of the Aztec diamond with the constrained boundary condition, which solved a conjecture posed by Stanley in 1994. In this paper, We give the formulae of the energy and the number of spanning trees of the Aztec diamond with the toroidal boundary condition. 相似文献
14.
Let G be a graph of order n and k a positive integer. A set of subgraphs H={H1,H2,…,Hk} is called a k-degenerated cycle partition (abbreviated to k-DCP) of G if H1,…,Hk are vertex disjoint subgraphs of G such that and for all i, 1≤i≤k, Hi is a cycle or K1 or K2. If, in addition, for all i, 1≤i≤k, Hi is a cycle or K1, then H is called a k-weak cycle partition (abbreviated to k-WCP) of G. It has been shown by Enomoto and Li that if |G|=n≥k and if the degree sum of any pair of nonadjacent vertices is at least n−k+1, then G has a k-DCP, except G≅C5 and k=2. We prove that if G is a graph of order n≥k+12 that has a k-DCP and if the degree sum of any pair of nonadjacent vertices is at least , then either G has a k-WCP or k=2 and G is a subgraph of K2∪Kn−2∪{e}, where e is an edge connecting V(K2) and V(Kn−2). By using this, we improve Enomoto and Li’s result for n≥max{k+12,10k−9}. 相似文献
15.
In this paper we address the Optimum Communication Spanning Tree Problem. We present a formulation that uses three index variables and we propose several families of inequalities, which can be used to reinforce the formulation. Preliminary computational experiments are very promising. 相似文献
16.
Zhi-Hong Chen 《Discrete Mathematics》2017,340(12):3104-3115
For a graph , let and let . We show that for a given number and given integers , and , if is a -connected claw-free graph of order with and its Ryjác?ek’s closure , and if where , then either is Hamiltonian or , the preimage of , can be contracted to a -edge-connected -free graph of order at most and without spanning closed trails. As applications, we prove the following for such graphs of order with sufficiently large:(i) If , , and for a given () , then either is Hamiltonian or where is a graph obtained from by replacing each of the degree 2 vertices by a (). When and , this proves a conjecture in Frydrych (2001).(ii) If , , and for a given () , then is Hamiltonian. These bounds on in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved and for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most are fixed for given , improvements to (i) or (ii) by increasing the value of are possible with the help of a computer. 相似文献
17.
A connected graph G is caterpillar-pure if each spanning tree of G is a caterpillar. The caterpillar-pure graphs are fully characterized. Loosely speaking they are strings or necklaces of so-called pearls, except for a number of small exceptional cases. An upper bound for the number of edges in terms of the order is given for caterpillar-pure graphs, and those which attain the upper bound are characterized. 相似文献
18.
In this paper, we prove that an m-connected graph G on n vertices has a spanning tree with at most k leaves (for k ≥ 2 and m ≥ 1) if every independent set of G with cardinality m + k contains at least one pair of vertices with degree sum at least n − k + 1. This is a common generalization of results due to Broersma and Tuinstra and to Win. 相似文献
19.
We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < ε < 1, there exists a constant c = c(d, ε) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − ε)n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and ε, then G has a copy of every tree T as above.
Research supported in part by a USA-Israeli BSF grant, by NSF grant CCR-0324906, by a Wolfensohn fund and by the State of
New Jersey.
Research supported in part by USA-Israel BSF Grant 2002-133, and by grants 64/01 and 526/05 from the Israel Science Foundation.
Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred
P. Sloan fellowship. 相似文献
20.
In 1990, Albertson, Berman, Hutchinson, and Thomassen proved a theorem which gives a minimum degree condition for the existence of a spanning tree with no vertices of degree 2. Such a spanning tree is called a homeomorphically irreducible spanning tree (HIST). In this paper, we prove that every graph of order () contains a HIST if for any nonadjacent vertices and . The degree sum condition is best possible. 相似文献