Maximizing $H$-colorings of connected graphs with fixed minimum degree

TL;DR

This paper investigates which connected graphs with fixed minimum degree maximize the number of $H$-colorings, revealing that for non-regular $H$, the complete bipartite graph $K_{ ext{min degree}, n- ext{min degree}}$ is optimal for large $n$, but this does not hold for all regular $H$.

Contribution

The paper establishes that for non-regular $H$, the extremal graph maximizing $H$-colorings is the complete bipartite graph $K_{ ext{min degree}, n- ext{min degree}}$, and shows exceptions for regular $H$.

Findings

For non-regular $H$, $K_{ ext{min degree}, n- ext{min degree}}$ uniquely maximizes $H$-colorings.

For large $n$, among $k$-connected graphs, $K_{k,n-k}$ maximizes the number of $H$-colorings.

Counterexamples exist for regular $H$, where other graphs surpass $K_{ ext{min degree}, n- ext{min degree}}$ in $H$-colorings.

Abstract

For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex graph with minimum degree $\delta$ maximizes the number of $H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the complete bipartite graph $K_{\delta,n-\delta}$ is the unique maximizer. As a corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$ is the unique $k$-connected graph that maximizes the number of $H$-colorings among all $k$-connected graphs. Finally, we show that this conclusion does not hold for all regular $H$ by exhibiting a connected $n$-vertex graph with minimum degree $\delta$ which has more $K_{q}$-colorings (for sufficiently large $q$ and $n$) than $K_{\delta,n-\delta}$.