The Independence Numbers of Weighted Graphs with Forbidden Cycles

Abstract

Let $G=(V,E,w)$ be a weighted graph. The independence number of a weighted graph G is the maximum w(S) taken over all independent sets S of G. By a well-known method, we can show that the independence number of G is at least $\sum _{v\in V} \frac{w(v)}{1+d(v)}$, where d(v) is the degree of v. In this paper, we consider the independence numbers of weighted graphs with forbidden cycles. For a graph G, the odd girth of G is the smallest length of an odd cycle in G. We show that if G is a weighted graph with odd girth $2m+1$ for $m\ge 3$, then the independence number of G is at least $c\left( \sum _{v}w(v)d(v)^{1/(m-2)}\right) ^{(m-2)/(m-1)}$, where c is a constant.

Supported in part by NSFC (12101156).