Maximizing H-colorings of a regular graph

TL;DR

This paper investigates the maximum number of H-colorings in regular graphs, proposing a conjecture, providing proofs for specific cases, and establishing asymptotic bounds based on structural properties of H.

Contribution

It introduces a conjecture on maximizing H-colorings in regular graphs, proves it for certain cases, and characterizes when different extremal graphs dominate.

Findings

Confirmed the conjecture for infinitely many triples (n, d, H)

Provided sharp estimates for hom(K_{d,d}, H) and hom(K_{d+1}, H)

Established asymptotic bounds for hom(G, H) in large degree limit

Abstract

For graphs $G$ and $H$, a {\em homomorphism} from $G$ to $H$, or {\em $H$-coloring} of $G$, is an adjacency preserving map from the vertex set of $G$ to the vertex set of $H$. Writing ${\rm hom}(G,H)$ 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 $$ {\rm hom}(G,H) \leq \max{{\rm hom}(K_{d,d},H)^{\frac{n}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{n}{d+1}}} $$ where $K_{d,d}$ is the complete bipartite graph with $d$ vertices in each partition class, and $K_{d+1}$ 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 ${\rm hom}(K_{d,d},H)^{n/2d}$. Here we exhibit infinitely many non-trivial triples $(n,d,H)$ for which the conjecture is true and for which the maximum is achieved by ${\rm hom}(K_{d+1},H)^{n/(d+1)}$. We also give sharp estimates for ${\rm hom}(K_{d,d},H)$ and ${\rm hom}(K_{d+1},H)$ in terms of some structural parameters of $H$. This allows us to characterize those $H$ for which ${\rm hom}(K_{d,d},H)^{1/2d}$ is eventually (for all sufficiently large $d$) larger than ${\rm hom}(K_{d+1},H)^{1/(d+1)}$ 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 $$ {\rm hom}(G,H)^{\frac{1}{|V(G)|}} \leq (1+o(1))\max{{\rm hom}(K_{d,d},H)^{\frac{1}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{1}{d+1}}} $$ where $o(1)\rightarrow 0$ as $d \rightarrow \infty$. More precise results are obtained in some special cases.