Maximum number of colourings. I. 4-chromatic graphs

TL;DR

This paper proves an upper bound on the number of k-colorings for connected 4-chromatic graphs, characterizes extremal graphs, and confirms a longstanding conjecture, with proof techniques applicable to related problems.

Contribution

It establishes the maximum number of k-colorings for connected 4-chromatic graphs and characterizes extremal graphs, confirming a conjecture and introducing novel proof methods.

Findings

Maximum number of k-colorings for connected 4-chromatic graphs is established.

Extremal graphs are characterized as those formed from K4 by adding leaves.

A new auxiliary result about list-chromatic polynomials solves a recent conjecture.

Abstract

It is proved that every connected graph $G$ on $n$ vertices with $\chi(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed from $K_4$ by repeatedly adding leaves. This confirms (a strengthening of) the $4$-chromatic case of a long-standing conjecture of Tomescu [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets etiquetes, C. R. Acad. Sci. Paris 273 (1971), 1124-1126]. Proof methods may be of independent interest. In particular, one of our auxiliary results about list-chromatic polynomials solves a recent conjecture of Brown, Erey, and Li.