On Selkow's Bound on the Independence Number of Graphs

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$d(v)=|N(v)|$denotetheneighborhoodandthedegreeofavertex$v\in V(G)$, respectively. However, Selkow’s original proof of this result is incorrect. We give a new probabilistic proof of Selkow’s bound here.