广义θ-链的区间边着色

Interval edge-coloring of the generalized θ-chain

如果图 G 的一个边着色用了 1,2,…,t 中的所有颜色,并且关联于 G 的同一个顶点的边上的颜色各不相同,且这些颜色构成了一个连续的整数区间,则称这个边着色是 G 的区间 t-着色。如果对某个正整数 t,G 有一个区间 t-着色,则称 G 是可区间着色的。所有可区间着色的图构成的集合记作 N。图 G 的亏度 def( G)是粘在 G 的顶点上使它可区间着色的悬挂边的最小数目,显然,G∈N 当且仅当 def( G)= 0。广义θ-链是把路 P =[v0,v1,…,v k]( k≥1)的每一条边 vi-1 vi( i = 1,2,…,k),用 mi≥2 条两两内部不交的( vi-1,vi)-路替换掉而得到的简单图,记作θm1,m2,…,mk。把广义θ-图亏度的结论进行推广,确定了θm1,m2,…,mk的亏度。

An edge-coloring of a graph G with colors 1,2,…,t is an interval t-coloring if all colors are used,and the colors of edges incident to each vertex of G are distinct and form a continuous interval of integer. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. The set of all interval colorable graphs is denoted by N. The deficiency def( G) of a graph G is the minimum number of pendant edges whose attachment to G makes it interval colorable. Obviously,G ∈N if and only if def( G)= 0. A generalized θ-chain,denoted by θm1,m2,…,mk,is a simple graph obtained by substituting each edge vi-1 vi of the path P =[v0,v1,…,v k]( k≥1) for mi≥2 pairwise internally disjoint ( vi-1,vi )-paths,where i = 1,2,…,k. In this paper,the conclu- sions of deficiency of generalized θ-graph are generalized,and the deficiency of the generalized θ-chain θm1,m2,…,mk is determined.