Long paths need not minimize $H$-colorings among trees

TL;DR

This paper explores the number of $H$-colorings of trees, showing that for certain symmetric target graphs, some non-path trees can have fewer colorings than paths, extending previous results to infinitely many sizes.

Contribution

The authors develop a new method to explicitly compare $H$-colorings of path-like trees and demonstrate the existence of trees with fewer colorings than paths for large sizes.

Findings

Existence of a target graph $H$ with trees having fewer $H$-colorings than paths for large $n$.

Extension of previous finite results to infinitely many tree sizes.

Introduction of a new enumeration strategy for homomorphisms from trees to symmetric graphs.

Abstract

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. By appropriate choice of $H$, these colorings can express, for instance, the independent sets or proper vertex colorings of $G$. Sidorenko proved that for any $H$, the $n$-vertex star admits at least as many $H$-colorings as any other $n$-vertex tree, but the minimization question remains open in general. For many graphs $H$, path graphs are among the trees with the fewest $H$-colorings, but work of Leontovich and subsequently Csikv\'ari and Lin shows that there is a graph $E_7$ on seven vertices and a target graph $H$ for which there are strictly fewer $H$-colorings of $E_7$ than of the path on seven vertices. We introduce a new strategy for enumerating homomorphisms from path-like trees to highly symmetric target graphs that allows us to make the previous observations completely explicit and extend them to infinitely many $n$ beyond $n=7$. In particular, we exhibit a target graph $H$ with the property that for each sufficiently large $n$, there is a tree $E_n$ on $n$ vertices that admits strictly fewer $H$-colorings than the path on $n$ vertices.