On Selkow's Bound on the Independence Number of Graphs

Abstract

For a graph $G$ with vertex set $V(G)$ and independence number $\alpha(G)$, S. M. Selkow (Discrete Mathematics, 132(1994)363--365) established the famous lower bound $\sum\limits_{v\in V(G)}\frac{1}{d(v)+1}(1+\max\{\frac{d(v)}{d(v)+1}-\sum\limits_{u\in N(v)}\frac{1}{d(u)+1},0 \})$ on $\alpha(G)$, where $N(v)$ and $d(v)=|N(v)|$ denote the neighborhood and the degree of a vertex $v\in V(G)$, respectively. However, Selkow's original proof of this result is incorrect. We give a new probabilistic proof of Selkow's bound here.