The Bipartite Swapping Trick on Graph Homomorphisms

TL;DR

This paper establishes an upper bound on the number of graph homomorphisms from regular graphs to a fixed graph, generalizing a conjecture on independent sets and applying to colorings and stable set polytopes.

Contribution

It introduces a bipartite swapping technique to bound homomorphisms, extending previous results and applying to various graph properties.

Findings

Derived an upper bound for homomorphisms from regular graphs to fixed graphs.

Generalized a conjecture of Alon and Kahn on independent sets.

Applied techniques to graph colorings and stable set polytopes.

Abstract

We provide an upper bound to the number of graph homomorphisms from $G$ to $H$, where $H$ is a fixed graph with certain properties, and $G$ varies over all $N$-vertex, $d$-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tetali, who studied the number of graph homomorphisms from $G$ to $H$ when $H$ is bipartite. We also apply our techniques to graph colorings and stable set polytopes.