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 \(\mathrm{IR}\) 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 \(\Delta \geq 1\), \(\mathrm{IR}(G) – \alpha(G) \leq \frac{\Delta -2}{2\Delta }n\). When \(\Delta = 3\), the result was conjectured by Rautenbach.