Counting proper colourings in 4-regular graphs via the Potts model

TL;DR

This paper establishes tight bounds on the number of proper q-colorings in 4-regular graphs using the antiferromagnetic Potts model, advancing conjectures in graph coloring and statistical physics.

Contribution

It proves the first case of a conjecture relating to bounds on the antiferromagnetic Potts model on 4-regular graphs and characterizes extremal graphs for proper colorings.

Findings

Tight bounds on the internal energy per particle for q ≥ 5.

Bounds on the number of proper q-colorings in 4-regular graphs.

Maximization of proper q-colorings by unions of K_{4,4}.

Abstract

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen, and Roberts on extensions of their methods, and implies tight bounds on the antiferromagnetic Potts partition function. The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$'s.