# Color Refinement, Homomorphisms, and Hypergraphs

## TL;DR

This paper generalizes the concept of graph color refinement to hypergraphs, establishing a homomorphism counting characterization for hypergraph similarity and relating hypergraph homomorphisms to vertex-colored graph homomorphisms.

## Contribution

It introduces a hypergraph color refinement generalization and proves a homomorphism-based characterization of hypergraph indistinguishability, extending known graph results.

## Key findings

- Hypergraph color refinement does not distinguish two hypergraphs if they have identical homomorphism counts from all connected Berge-acyclic hypergraphs.
- Homomorphisms of hypergraphs relate to those of colored incidence graphs, reducing hypergraph problems to graph problems.
- The approach extends the Tree Theorem from graphs to hypergraphs.

## Abstract

Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish two graphs G and H if and only if, for every tree T, the number of homomorphisms Hom(T,G) from T to G is equal to the corresponding number Hom(T,H) from T to H (Dell, Grohe, Rattan 2018). We show how this approach transfers to hypergraphs by introducing a generalization of color refinement. We prove that it does not distinguish two hypergraphs G and H if and only if, for every connected Berge-acyclic hypergraph B, we have Hom(B,G) = Hom(B,H). To this end, we show how homomorphisms of hypergraphs and of a colored variant of their incidence graphs are related to each other. This reduces the above statement to one about vertex-colored graphs.