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1.
. In this work we consider finite undirected simple graphs. If G=(V,E) is a graph we denote by α(G) the stability number of G. For any vertex x let N[x] be the union of x and the neighborhood N(x). For each pair of vertices ab of G we associate the set J(a,b) as follows. J(a,b)={u∈N[a]∩N[b]∣N(u)⊆N[a]∪N[b]}. Given a graph G, its partially squareG
* is the graph obtained by adding an edge uv for each pair u,v of vertices of G at distance 2 whenever J(u,v) is not empty. In the case G is a claw-free graph, G
* is equal to G
2.
If G is k-connected, we cover the vertices of G by at most ⌈α(G
*)/k⌉ cycles, where α(G
*) is the stability number of the partially square graph of G. On the other hand we consider in G
* conditions on the sum of the degrees. Let G be any 2-connected graph and t be any integer (t≥2). If ∑
x
∈
S
deg
G
(x)≥|G|, for every t-stable set S⊆V(G) of G
* then the vertex set of G can be covered with t−1 cycles. Different corollaries on covering by paths are given.
Received: January 22, 1997 Final version received: February 15, 2000 相似文献
2.
徐新萍 《数学的实践与认识》2009,39(10)
设G是一个图,G的部分平方图G*满足V(G*)=V(G),E(G*)=E(G)∪{uv:uv■E(G),且J(u,v)≠■},这里J(u,v)={w∈N(u)∩N(v):N(w)■N[u]∪N[v]}.利用插点方法,证明了如下结果:设G是k-连通图(k2),b是整数,0min {k,(2b-1+k)/2}(n(Y)-1),则G是哈密尔顿图.同时给出图是1-哈密尔顿的和哈密尔顿连通的相关结果. 相似文献
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4.
We study new posets Q obtained by removing from a geometric lattice L ofa biased graph certain flats indexed by a simplicial complex
. (One example of L is the lattice of flats of thevector matroid of a root system B
n
.) We study the structureand compute the characteristic polynomial of Q. With certainchoices of L and
, including ones for which Q is alattice interpolating between those of B
n
and D
n
, we observe curious relationships among the roots of thecharacteristic polynomials of Q, L, and
. 相似文献
5.
A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, a complete classification is given of tetravalent vertex-transitive non-Cayley graphs of order \(2p^2\) for any prime p. 相似文献
6.
对于Hamiltonian极小和具有共形Maslov形式的Lagrangian图,建立了Bernstein型的定理,推广了已知关于极小Lagrangian子流形的一些结果. 相似文献
7.
Siberian Mathematical Journal - Considering the partially commutative groups of varieties which contain the variety of all nilpotent groups of class 2 as well as the... 相似文献
8.
作为部分线性模型和变系数模型的推广,部分线性变系数模型以其良好的适应性和稳健性受到了广泛的关注。本文基于函数的局部线性拟合,给出部分线性变系数模型的另一种轮廓(profile)最小二乘估计的方法,并从理论上证实了所得估计量具有良好的渐近性质,最后给出了估计方法的实例分析。 相似文献
9.
Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs I: Latin Square Constructions 下载免费PDF全文
In an earlier paper the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part I, we explore a connection between orthogonal latin squares and embeddings. A product construction is presented for building pairs of orthogonal latin squares such that one member of the pair has a certain hamiltonian property. These hamiltonian squares are then used to construct embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that and for every prime p. Moreover, it is shown that the latin square construction utilized to get hamilton cycle embeddings of can also be used to obtain triangulations of . Part II of this series covers the case for every prime p and applies these embeddings to obtain some genus results. 相似文献
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Sanming Zhou 《Monatshefte für Mathematik》2003,139(1):69-81
With any G-symmetric graph Γ admitting a nontrivial G-invariant partition , we may associate a natural “cross-sectional” geometry, namely the 1-design in which for and if and only if α is adjacent to at least one vertex in C, where and is the neighbourhood of B in the quotient graph of Γ with respect to . In a vast number of cases, the dual 1-design of contains no repeated blocks, that is, distinct vertices of B are incident in with distinct subsets of blocks of . The purpose of this paper is to give a general construction of such graphs, and then prove that it produces all of them.
In particular, we show that such graphs can be reconstructed from and the induced action of G on . The construction reveals a close connection between such graphs and certain G-point-transitive and G-block-transitive 1-designs. By using this construction we give a characterization of G-symmetric graphs such that there is at most one edge between any two blocks of . This leads to, in a subsequent paper, a construction of G-symmetric graphs such that and each is incident in with vertices of B.
The work was supported by a discovery-project grant from the Australian Research Council.
Received April 24, 2001; in revised form October 9, 2002
Published online May 9, 2003 相似文献
14.
16.
Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai (Combinatorics and Graph Theory, vol 95, World Scientific, Singapore, pp 53–69; Conjecture 8.6 of 1995) conjectured that every 3-edge connected and essentially 6-edge connected graph is collapsible. Denote D 3(G) the set of vertices of degree 3 of graph G. For ${e = uv \in E(G)}$ , define d(e) = d(u) + d(v) ? 2 the edge degree of e, and ${\xi(G) = \min\{d(e) : e \in E(G)\}}$ . Denote by λ m (G) the m-restricted edge-connectivity of G. In this paper, we prove that a 3-edge-connected graph with ${\xi(G)\geq7}$ , and ${\lambda^3(G)\geq7}$ is collapsible; a 3-edge-connected simple graph with ${\xi(G)\geq7}$ , and ${\lambda^3(G)\geq6}$ is collapsible; a 3-edge-connected graph with ${\xi(G)\geq6}$ , ${\lambda^2(G)\geq4}$ , and ${\lambda^3(G)\geq6}$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected simple graph with ${\xi(G)\geq6}$ , and ${\lambda^3(G)\geq5}$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected graph with ${\xi(G)\geq5}$ , and ${\lambda^2(G)\geq4}$ with at most 9 vertices of degree 3 is collapsible. As a corollary, we show that a 4-connected line graph L(G) with minimum degree at least 5 and ${|D_3(G)|\leq9}$ is Hamiltonian. 相似文献
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Zhi-Hong Chen 《Graphs and Combinatorics》2016,32(6):2267-2273
A graph G is hypohamiltonian if it is not Hamiltonian but for each \(v\in V(G)\), the graph \(G-v\) is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset \(R\subseteq V(G)\), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173–181, 1990] and in [J. Combin. Theory, Ser B 66:123–139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291–304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Université Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013. 相似文献
20.
Michele Conforti Gérard Cornuéjols Grigor Gasparyan Kristina Vušković 《Combinatorica》2002,22(1):19-33
We prove a theorem about cutsets in partitionable graphs that generalizes earlier results on amalgams, 2-amalgams and homogeneous
pairs.
Received December 13, 1999
RID="*"
ID="*" This work was supported in part by the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, and
by NSF grants DMI-0098427 and DMI-9802773 and ONR grant N00014-97-1-0196. 相似文献