**Journal of Graph Theory**

# Maximizing *H*-Colorings of a Regular Graph

Contract grant sponsor: National Security Agency; Contract grant number: H98230-10-1-0364.

## Abstract

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 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

where is the complete bipartite graph with *d* vertices in each partition class, and is the complete graph on vertices.Results of Zhao confirm this conjecture for some choices of *H* for which the maximum is achieved by . Here, we exhibit for the first time infinitely many nontrivial triples for which the conjecture is true and for which the maximum is achieved by .We also give sharp estimates for and in terms of some structural parameters of *H*. This allows us to characterize those *H* for which is eventually (for all sufficiently large *d*) larger than 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

where as . More precise results are obtained in some special cases.