Counterexamples to two conjectures on mean color numbers of graphs

TL;DR

This paper constructs infinite counterexamples to two conjectures regarding the mean color number of graphs, challenging previous assumptions about how edge deletions and vertex removal affect this graph invariant.

Contribution

The paper provides the first known infinite family of counterexamples to two conjectures on mean color numbers, disproving their general validity.

Findings

Counterexamples invalidate the conjectures.

Mean color number behavior is more complex than previously thought.

The results impact understanding of graph coloring properties.

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.