A Note on the Difference Between the Upper Irredundance and Independence Numbers of a Graph

Abstract

Let $G=(V,E)$ be a simple graph. Let $\alpha$ and $\mathrm{I}\mathrm{R}$ be the independence number and upper irredundance number of $G$, respectively. In this paper, we prove that for any graph $G$ of order $n$ with maximum degree $\mathrm{\Delta }\ge 1$, $\mathrm{I}\mathrm{R}(G)–\alpha (G)\le \frac{\mathrm{\Delta }-2}{2\mathrm{\Delta }}n$. When $\mathrm{\Delta }=3$, the result was conjectured by Rautenbach.