Graph Operations and Upper Bounds on Graph Homomorphism Counts

TL;DR

This paper constructs counterexamples to a conjecture on upper bounds of graph homomorphism counts and identifies new classes of graphs where these bounds hold, including a case confirming Galvin's conjecture for the Widom-Rowlinson graph.

Contribution

It provides counterexamples to Galvin's conjecture and establishes new infinite families of graphs satisfying the conjectured bounds, using graph tensor product and exponentiation techniques.

Findings

Counterexamples to Galvin's conjecture are constructed.

New classes of graphs are identified where the bounds hold.

The conjecture is verified for the Widom-Rowlinson graph.

Abstract

We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any $n$-vertex, $d$-regular graph $G$ and any graph $H$ (possibly with loops), \[\hom(G,H) \leq \max\left\lbrace\hom(K_{d,d}, H)^{\frac{n}{2d}}, \hom(K_{d+1},H)^{\frac{n}{d+1}}\right\rbrace,\] where $\hom(G,H)$ is the number of homomorphisms from $G$ to $H$. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of $H$ for which the bound stated above on $\hom(G,H)$ holds for all $n$-vertex, $d$-regular $G$. In particular we show that if $H_{\rm WR}$ is the complete looped path on three vertices, also known as the Widom-Rowlinson graph, then $$ {\hom}(G,H_{\rm WR}) \leq {\hom}(K_{d+1},H_{\rm WR})^\frac{n}{d+1} $$ for all $n$-vertex, $d$-regular $G$. This verifies a conjecture of Galvin.