%0 Journal Article
%T On the Widom-Rowlinson Occupancy Fraction in Regular Graphs
%A Emma Cohen
%A Will Perkins
%A Prasad Tetali
%J Mathematics
%D 2015
%I arXiv
%X We consider the Widom-Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction: the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d+1 vertices, $K_{d+1}$'s. As a corollary we find that $K_{d+1}$ also maximizes the normalized partition function of the Widom-Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph $H_{WR}$, a path on three vertices with a self-loop on each vertex, is maximized by $K_{d+1}$. This proves a conjecture of Galvin.
%U http://arxiv.org/abs/1512.06398v1