超级-λ'无三角图的度和充分条件

Degree and Sum Conditions for Triangle-free Graphs to be Super Restricted Edge-connected

设S是连通图G的一个边割。若G-S不包含孤立点,则称S是G的一个限制边割。如果图G的每个最小限制边割恰好分离出图G的一条边,则称图G是超级限制边连通的,简称超级-λ'的。设G是一个阶n≥4的连通无三角图。本文证明了若G中任意满足dist(u,v)=2的点对u,v∈V(G)有d(u)+d(v)≥2[n+2/4]+3,则G是超级-λ'的。最后,举例说明该结论是最好的。

An edge cut S of a connected graph G is called as a restricted edge cut if G-S contains no isolated vertices. A graph is to be super restricted edge-connected for short super-λ＇,if every minimum restricted edge cut isolates an edge. In this paper,we study the degree sum conditions for triangle-free graphs to be super restricted edge connectivity,and prove that： Let G be a connected triangle-free graph of order. If d（ u） ＋ d（ v） ≥2[（n ＋2）/4]＋ 3 for each pair vertices u,v∈V（ G） with dist（ u,v） = 2,then G is super-λ＇. Moreover,the result is demonstrated to be the best possible.