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

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.

17 pages, 2 figures

References in corpus (4)

• On the average size of independent sets in triangle-free graphs
• Extremal regular graphs: independent sets and graph homomorphisms
• Extremes of the internal energy of the Potts model on cubic graphs
• Tight bounds on the coefficients of partition functions via stability

Cited by in corpus (1)

• The number of independent sets in an irregular graph