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

LetHWR be the path on 3 vertices with a loop at each vertex. D. Galvin [4, 5] conjectured, and E. Cohen, W. Perkins and P. Tetali [2] proved that for any d-regular simple graph G on n vertices we have hom(G,HWR) ≤ hom(Kd+1, HWR). In this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali [2]. 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) ≤ hom(Kd+1, H). 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.