Maximizing the number of q-colorings

TL;DR

This paper determines the structure of graphs that maximize the number of proper q-colorings for various parameters, extending previous results and confirming longstanding conjectures.

Contribution

It introduces a new approach to identify extremal graphs maximizing q-colorings for broad parameter ranges, including large m and specific q values.

Findings

For q >= 4 and large m, extremal graphs are complete bipartite minus a star edges plus isolated vertices.

Confirmed Lazebnik's conjecture for q=3 and large m, describing the optimal graph structure.

Provided a framework to solve the maximization problem for many nontrivial parameter ranges.

Abstract

Let P_G(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing P_G(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing P_G(q) over various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach which enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q >= 4 and sufficiently large m < \kappa_q n^2 where \kappa_q is approximately 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m <= n^2/4, confirming (in a stronger form) a conjecture of Lazebnik from 1989.