# On Selkow's Bound on the Independence Number of Graphs

## TL;DR

This paper provides a new probabilistic proof for Selkow's lower bound on the independence number of graphs, correcting the original proof which was found to be incorrect.

## Contribution

It introduces a valid probabilistic proof for Selkow's bound, enhancing the theoretical understanding of graph independence numbers.

## Key findings

- Established a correct probabilistic proof of Selkow's bound
- Confirmed the validity of the lower bound on independence number
- Clarified the conditions under which the bound holds

## 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.