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 paper 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) ≤ max { hom(Kd,d, H) n 2d , hom(Kd+1, H) n d+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 non-trivial 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 non-trivial 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)) max { hom(Kd,d, H) 1 2d ,hom(Kd+1, H) 1 d+1 } where o(1)→ 0 as d→∞. More precise results are obtained in some special cases. ∗Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame IN 46556; [email protected]. Research supported in part by National Security Agency grant H98230-10-1-0364.