Maximizing H-Colorings of a Regular Graph

Authors


  • 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 math formula 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

display math

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

display math

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

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