On the Widom-Rowlinson Occupancy Fraction in Regular Graphs

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 maximises the normalised partition function of the Widom-Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalised number of homomorphisms from any d-regular graph $G$ to the graph $H_{WR}$, a path on three vertices with a loop on each vertex, is maximised by $K_{d+1}$. This proves a conjecture of Galvin.