# Counting Homomorphisms Modulo a Prime Number

## TL;DR

This paper classifies the computational complexity of counting graph homomorphisms modulo a prime number p, extending previous results to a broader class of square-free graphs and showing the problem is #_p P-hard unless H is a star.

## Contribution

It extends the complexity classification of #_p GraphHom(H) to all square-free graphs H that are not stars, generalizing prior work for specific cases.

## Key findings

- The problem is #_p P-hard for all square-free graphs H that are not stars.
- Previous results were limited to trees and specific graph classes.
- Complete classification of the complexity for square-free graphs H.

## Abstract

Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is #GraphHom(H) --- the problem of finding the number of homomorphisms from a given graph G to the graph H. Not only the complexity of this basic problem is known, but also of its many variants for digraphs, more general relational structures, graphs with weights, and others. In this paper we consider a modification of #GraphHom(H), the #_p GraphHom(H) problem, p a prime number: Given a graph G, find the number of homomorphisms from G to H modulo p. In a series of papers Faben and Jerrum, and Goebel et al. determined the complexity of #_2 GraphHom(H) in the case H (or, in fact, a certain graph derived from H) is square-free, that is, does not contain a 4-cycle. Also, Goebel et al. found the complexity of #_p GraphHom(H) for an arbitrary prime p when H is a tree. Here we extend the above result to show that the #_p GraphHom(H) problem is #_p P-hard whenever the derived graph associated with H is square-free and is not a star, which completely classifies the complexity of #_p GraphHom(H) for square-free graphs H.