Hoffman-London graphs: When paths minimize $H$-colorings among trees

TL;DR

This paper introduces a new method using automorphisms of target graphs to identify trees that minimize the number of $H$-colorings, revealing Hoffman-London graphs and applying the approach to various graph families.

Contribution

It develops a novel automorphism-based technique to determine Hoffman-London graphs that minimize $H$-colorings among trees, including a full characterization for small graphs.

Findings

Paths minimize $ ext{hom}(T,H)$ for Hoffman-London graphs.

Identified families of Hoffman-London graphs, including loop threshold graphs.

Characterized minimizing trees for all graphs with up to three vertices.

Abstract

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. The number of $H$-colorings of $G$, $\hom(G,H)$, has been studied for many classes of $G$ and $H$. In particular, extremal questions of maximizing and minimizing $\hom(G,H)$ have been considered when $H$ is a clique or $G$ is a tree. In this paper, we develop a new technique using automorphisms of $H$ to show that $\hom(T,H)$ is minimized by paths as $T$ varies over trees on a fixed number of vertices. We introduce the term Hoffman-London to refer to graphs that are minimal in this sense. In particular, we define an automorphic similarity matrix which is used to compute $\hom(T,H)$ and give matrix conditions under which $H$ is Hoffman-London. We then apply this technique to identify several families of graphs that are Hoffman-London, including loop threshold graphs and some with applications in statistical physics (e.g. the Widom-Rowlinson model). By combining our approach with a few other observations, we fully characterize the minimizing trees for all graphs $H$ on three or fewer vertices.