共查询到19条相似文献,搜索用时 109 毫秒
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Let G be a graph. An independent set Y in G is called an essential independent set (or essential set for simplicity) if there is {Y1, Y2} 包含于Y such that dist (y1,y2)=2. In this paper, we use the technique of the vertex insertion on l-connected (l=k or k 1, k≥2) graphs to provide a unified proof for G to be hamiltonian, or hamiltonian-connected. The sufficient conditions are expressed an inequality on ∑i=1 K|N(Yi)| b|N(y0)| and n(Y) for each essential set Y={y0,y1,…,yk}, where b (1≤b≤k)is an integer,Yi={yi,yi-1,…,yi-(b-1}包含于Y\{y0} for i属于V(G):dist(v,Y)≤2}|. 相似文献
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研究子图的度和图的哈密尔顿性的关系,证明图~$G$ 是一个~$n$ 阶~3-\,连通无爪图且最小度~$\delta(G)\geq4$, 如果图~$G$ 中任意两个分别同构于~$P_4$, $K_1$ 的不相邻子图~$H_1$, $H_2$ 满足~$d(H_1)+d(H_2)\geq n$, 则图~$G$ 是哈密尔顿连通. 相似文献
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在文献[3]中介绍了一个新的图类-P3-支配图.这个图类包含所有的拟无爪图,因此也包含所有的无爪图.在本文中,我们证明了每一个点数至少是3的三角形连通的P3-支配图是哈密尔顿的,但有一个例外图K1,1,3.同时,我们也证明了k-连通的(k≥2)的P3-支配图是哈密尔顿的,如果an(G)≤k,但有两个例外图K1,1,3 and K2,3. 相似文献
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设D是一个有向伪图,如果对于任意两个点u和v,D有一条生成(u,v)-路或一条生成(v,u)-路,则D是弱哈密尔顿连通的;若既存在一条生成(u,v)-路又存在一条生成(v,u)-路,则D是强哈密尔顿连通的.一个有向伪图D的线图L(D)是D的弧集作为其点集,对于任意两个点a,b∈A(D),(a,b)是L(D)的弧当且仅当存在D中的点u,v,w满足a=(u,v)并且b=(v,w).本文刻画了两类有向伪图T及T’,使得L(D)是弱哈密尔顿连通的当且仅当D∈T,并且L(D)是强哈密尔顿连通的当且仅当D∈T’. 相似文献
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邻接树图是哈密尔顿图猜想的一个等价命题 总被引:1,自引:0,他引:1
本文给出了简单图的邻接树图是哈密尔顿图”猜想的等价命题,阐明只需证明该猜想对2-连通图成立即可,另外,我们给出了该猜想一种特殊情形的构造性证明。 相似文献
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Let Γ be a connected simple graph, let V(Γ) and E(Γ) denote the vertex-set and the edge-set of Γ, respectively, and let n=|V(Γ)|. For 1≤i≤n, let ei be the element of elementary abelian group which has 1 in the ith coordinate, and 0 in all other coordinates. Assume that V(Γ)={ei∣1≤i≤n}. We define a set Ω by Ω={ei+ej∣{ei,ej}∈E(Γ)}, and let CayΓ denote the Cayley graph over with respect to Ω. It turns out that CayΓ contains Γ as an isometric subgraph. In this paper, the relations between the spectra of Γ and CayΓ are discussed. Some conditions on the existence of Hamilton paths and cycles in Γ are obtained. 相似文献
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For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−cD if and only if for every n≥cD+3, has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result. 相似文献
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We study connectivity, Hamilton path and Hamilton cycle decomposition, 4-edge and 3-vertex coloring for geometric graphs arising from pseudoline (affine or projective) and pseudocircle (spherical) arrangements. While arrangements as geometric objects are well studied in discrete and computational geometry, their graph theoretical properties seem to have received little attention so far. In this paper we show that they provide well-structured examples of families of planar and projective-planar graphs with very interesting properties. Most prominently, spherical arrangements admit decompositions into two Hamilton cycles; this is a new addition to the relatively few families of 4-regular graphs that are known to have Hamiltonian decompositions. Other classes of arrangements have interesting properties as well: 4-connectivity, 3-vertex coloring or Hamilton paths and cycles. We show a number of negative results as well: there are projective arrangements which cannot be 3-vertex colored. A number of conjectures and open questions accompany our results. 相似文献
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In this paper we study a graph operation which produces what we call the “vertex envelope” G∗V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope. 相似文献
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The longest path problem is a well-known NP-hard problem and so far it has been solved polynomially only for a few classes of graphs. In this paper, we give a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph. 相似文献
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Let cl(G) denote Ryjá?ek's closure of a claw‐free graph G. In this article, we prove the following result. Let G be a 4‐connected claw‐free graph. Assume that G[NG(T)] is cyclically 3‐connected if T is a maximal K3 in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.'s theorem [J Graph Theory 48(4) (2005), 267–276] and Pfender's theorem [J Graph Theory 49(4) (2005), 262–272]. © 2011 Wiley Periodicals, Inc. J Graph Theory 相似文献
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We consider the Hamiltonian cycle problem embedded in singularly perturbed (controlled) Markov chains. We also consider a functional on the space of stationary policies of the process that consists of the (1,1)‐entry of the fundamental matrices of the Markov chains induced by the same policies. In particular, we focus on the subset of these policies that induce doubly stochastic probability transition matrices, which we refer to as the “doubly stochastic policies.” We show that when the perturbation parameter ? is sufficiently small the minimum of this functional over the space of the doubly stochastic policies is attained very close to a Hamiltonian cycle, provided that the graph is Hamiltonian. We also derive precise analytical expressions for the elements of the fundamental matrix that lend themselves to probabilistic interpretation as well as asymptotic expressions for the first diagonal element, for a variety of deterministic policies that are of special interest, including those that correspond to Hamiltonian cycles. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004 相似文献
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Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5-E(C4), where C4 is a cycle of length 4 in K5. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected. 相似文献
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In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs. 相似文献