The binding number of a graph and its cliques

Abstract

We consider the binding numbers of $K_r$-free graphs, and improve the upper bounds on the binding number which force a graph to contain a clique of order $r$. For the case $r=4$, we provide a construction for $K_4$-free graphs which have a larger binding number than the previously known constructions. This leads to a counterexample to a conjecture by Caro regarding the neighborhoods of independent sets.