The extremality of 2-partite Tur\'an graphs with respect to the number of colorings

TL;DR

This paper proves that for large graphs with a fixed number of edges, the 2-partite Turán graph maximizes the number of proper q-colorings when q is an odd integer at least 5, confirming a conjecture for these parameters.

Contribution

It establishes that the 2-partite Turán graph uniquely maximizes the number of q-colorings among graphs with the same size and edge count for large n and odd q ≥ 5.

Findings

T_2(n) maximizes q-colorings for large n and odd q ≥ 5

Equality holds only for the Turán graph itself

The proof reduces the problem to a quadratic programming formulation

Abstract

We consider a problem proposed by Linial and Wilf to determine the structure of graphs that allows the maximum number of $q$-colorings among graphs with $n$ vertices and $m$ edges. Let $T_r(n)$ denote the Tur\'{a}n graph - the complete $r$-partite graph on $n$ vertices with partition sizes as equal as possible. We prove that for all odd integers $q\geq 5$ and sufficiently large $n$, the Tur\'{a}n graph $T_2(n)$ has at least as many $q$-colorings as any other graph $G$ with the same number of vertices and edges as $T_2(n)$, with equality holding if and only if $G=T_2(n)$. Our proof builds on methods by Norine and by Loh, Pikhurko, and Sudakov, which reduces the problem to a quadratic program.