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

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.

Chromatic roots and limits of dense graphs

In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.