A sufficient condition for a graph to be hamiltonian

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.     

A sufficient condition for a regular graph to be class 1

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.     

平面图3色可染的一个充分条件

Steinberg猜想既没有4-圈又没有5-圈的平面图是3色可染的. Xu, Borodin等人各自独立地证明了既没有相邻三角形又没有5-和7-圈的平面图是3 色可染的. 作为这一结果的推论, 没有4-, 5-和7-圈的平面图是3色可染的. 本文证明一个比此推论更接近Steinberg猜想的结果, 设G是一个既没有4-圈又没有5-圈的平面图, 若对每一个k∈{3, 6, 7}, G都不含(k, 7)-弦, 则G是3色可染的, 这里的(k, 7)-弦是指长度为7+k-2的圈的一条弦, 它的两个端点将圈分成两条路, 一条路的长度为6, 另一条路的长度为k-1.     

A sufficient condition for a bipartite graph to have a k-factor

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.     

最大度为6的平面图为第一类的一个新充分条件

本文证明了:若一个平面图G不含带弦的6-圈,则G是第一类的.这部分地证实了Vizing的关于平面图边染色的一个猜想.     

A sufficient condition for a polynomial to be stable

A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n∈N we find the smallest possible constant dn>0 such that if the coefficients of F(z)=a₀+a₁z+?+anzn are positive and satisfy the inequalities akak+1>dnak−1ak+2 for k=1,2,…,n−2, then F(z) is Hurwitz.     

A sufficient condition for a digraph to be kernel-perfect

Galeana-Sanchez and Neumann-Lara proved that a sufficient condition for a digraph to have a kernel (i.e., an absorbent independent set) is the following: (P) every odd directed cycle possesses at least two directed chords whose terminal endpoints are consecutive on the cycle. Here it is proved that (P) is satisfied by those digraphs having these two properties: (i) the reversal of every 3-circuit is present, and (ii) every odd directed cycle v₁… v_2n+1 V₁ has two chords of the form (v_i, v_i+2). This is stronger than a result of Galeana-Sanchez.     

A sufficient condition for a monoid to be a group

I give a sufficient condition for Monoid (G,?) to become a group when one of its elements is deleated. The entire thing has been put in the form of a theorem with its proof and it has been discussed broadly.     

A sufficient condition for a group automorphism to be algebraic

Translated from Algebra i Logika, Vol. 28, No. 1, pp. 117–119, January–February, 1989.     

A sufficient condition for graphs to be weaklyk-linked

We consider graphs, which are finite, undirected, without loops and in which multiple edges are possible. For each natural numberk letg(k) be the smallest natural numbern, so that the following holds:LetG be ann-edge-connected graph and lets ₁,...,s _k,t ₁,...,t _k be vertices ofG. Then for everyi {1,..., k} there existsa pathP _i froms _i tot _i, so thatP ₁,...,P _k are pairwise edge-disjoint. We prove      

A sufficient condition for a graph to contain three disjoint 1-factors

Ore derived a sufficient condition for a graph to contain a Hamiltonian cycle. We obtain a sufficient condition, similar to Ore's condition, for a graph to contain a Hamiltonian cycle and a 1-factor which are edge disjoint.     

A sufficient condition for a graph to have [a,b]-factors

We give a sufficient condition by using neighborhoods for a graph to have [a, b]-factors.Dedicated to Professor Tuyosi Oyama on his 60th Birthday