Maximizing H-Colorings of a Regular Graph

For graphs G and H, a homomorphism from G to H, or H‐coloring of G, is an adjacency preserving map from the vertex set of G to the vertex set of H. Our concern in this article is the maximum number of H‐colorings admitted by an n‐vertex, d‐regular graph, for each H. Specifically, writing hom(G,H) for the number of H‐colorings admitted by G, we conjecture that for any simple finite graph H (perhaps with loops) and any simple finite n‐vertex, d‐regular, loopless graph G, we have hom(G,H)≤maxhom(Kd,d,H)n2d,hom(Kd+1,H)nd+1,where Kd,d is the complete bipartite graph with d vertices in each partition class, and Kd+1 is the complete graph on d+1 vertices.Results of Zhao confirm this conjecture for some choices of H for which the maximum is achieved by hom(Kd,d,H)n/2d. Here, we exhibit for the first time infinitely many nontrivial triples (n,d,H) for which the conjecture is true and for which the maximum is achieved by hom(Kd+1,H)n/(d+1).We also give sharp estimates for hom(Kd,d,H) and hom(Kd+1,H) in terms of some structural parameters of H. This allows us to characterize those H for which hom(Kd,d,H)1/2d is eventually (for all sufficiently large d) larger than hom(Kd+1,H)1/(d+1) and those for which it is eventually smaller, and to show that this dichotomy covers all nontrivial H. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed H, for all d‐regular G, we have hom(G,H)1|V(G)|≤(1+o(1))maxhom(Kd,d,H)12d,hom(Kd+1,H)1d+1,where o(1)→0 as d→∞. More precise results are obtained in some special cases.