A proof of Tomescu's graph coloring conjecture

TL;DR

This paper proves Tomescu's conjecture that connected graphs with a given chromatic number have a maximum number of proper colorings, characterizing the extremal graphs for all relevant values of k.

Contribution

It completes the proof of Tomescu's conjecture for all k ≥ 4 and characterizes the extremal graphs achieving equality.

Findings

Confirmed the conjecture for all k ≥ 4.

Identified extremal graphs as k-cliques with attached trees.

Established the uniqueness of extremal graphs.

Abstract

In 1971, Tomescu conjectured that every connected graph $G$ on $n$ vertices with chromatic number $k\geq4$ has at most $k!(k-1)^{n-k}$ proper $k$-colorings. Recently, Knox and Mohar proved Tomescu's conjecture for $k=4$ and $k=5$. In this paper, we complete the proof of Tomescu's conjecture for all $k\ge 4$, and show that equality occurs if and only if $G$ is a $k$-clique with trees attached to each vertex.