Counterexamples to two conjectures on mean color numbers of graphs

Abstract

The mean color number of an $n$-vertex graph $G$, denoted by $\mu(G)$, is the average number of colors used in all proper $n$-colorings of $G$. For any graph $G$ and a vertex $w$ in $G$, Dong (2003) conjectured that if $H$ is a graph obtained from a graph $G$ by deleting all but one of the edges which are incident to $w$, then $\mu(G)\geq \mu(H)$; and also conjectured that $\mu(G)\geq \mu((G-w)\cup K_1)$. We prove that there is an infinite family of counterexamples to these two conjectures.