The Widom-Rowlinson model, the hard-core model and the extremality of the complete graph

TL;DR

This paper provides a concise proof of a conjecture relating homomorphisms of regular graphs to the Widom-Rowlinson and hard-core models, extending the inequality to new classes of graphs such as even cycles and paths.

Contribution

The authors offer a simplified proof of a key conjecture and establish the inequality for additional graph classes, broadening the understanding of homomorphism bounds.

Findings

Proved the conjecture relating homomorphisms to the Widom-Rowlinson model.

Extended the inequality to paths and even cycles with loops.

Established a bijection between the Widom-Rowlinson and hard-core models.

Abstract

Let $H_{\mathrm{WR}}$ be the path on $3$ vertices with a loop at each vertex. D. Galvin conjectured, and E. Cohen, W. Perkins and P. Tetali proved that for any $d$-regular simple graph $G$ on $n$ vertices we have $$\hom(G,H_{\mathrm{WR}})\leq \hom(K_{d+1},H_{\mathrm{WR}})^{n/(d+1)}.$$ In this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali. Our main tool is a simple bijection between the Widom-Rowlinson model and the hard-core model on another graph. We also give a large class of graphs $H$ for which we have $$\hom(G,H)\leq \hom(K_{d+1},H)^{n/(d+1)}.$$ In particular, we show that the above inequality holds if $H$ is a path or a cycle of even length at least $6$ with loops at every vertex.