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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. 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 HWR , a path on three vertices with a loop on each vertex, is maximized by K d+1. This proves a conjecture of Galvin.