Lower bounds on the independence number of a graph in terms of degrees

Abstract

Given an integer $\Delta \ge 3$, let ${\cal G}_{\Delta }$ be the set of connected graphs $G\neq K_{\Delta +1}$ with maximum degree $\Delta $ and, for $i=1,\cdots, \Delta $, let $V_i(G)$ be the set of vertices of $G$ of degree $i$. \\ We prove that $\sum\limits_{i=1}^\Delta c_i|V_i(G)|$ is a lower bound on the independence number $\alpha(G)$ of $G\in {\cal G}_\Delta$, where $c_\Delta=\frac{1}{\Delta}$ and $ic_{i}=1-c_{i+1}$ for $i=1,\cdots,\Delta-1$. Moreover, if $\varepsilon >0$ and $j\in \{1,\cdots, \Delta\}$, then the inequality $\alpha(G)\ge \varepsilon|V_j(G)|+\sum\limits_{i=1}^\Delta c_i|V_i(G)|$ does not hold for infinitely many graphs $G\in {\cal G}_\Delta$. We also show that an independent set $I\subset V(G)$ of $G\in {\cal G}_\Delta$ such that $|I|\ge \sum\limits_{i=1}^\Delta c_i|V_i(G)|$ can be found in polynomial time.