The Bipartite Swapping Trick on Graph Homomorphisms

We provide an upper bound to the number of graph homomorphisms from G to H, where H is a fixed graph with certain properties, and G varies over all N -vertex, d-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tetali, who studied the number of graph homomorphisms from G to H when G is bipartite. We also apply our techniques to graph colorings and stable set polytopes.