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 inline image 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

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

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

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