A topological proof of the Hell-Ne\v{s}et\v{r}il dichotomy

TL;DR

This paper offers a novel topological combinatorics proof of the Hell-Nešetřil dichotomy theorem, which classifies the computational complexity of graph homomorphism problems based on the target graph.

Contribution

The authors present a new proof of the Hell-Nešetřil theorem using topological combinatorics, connecting it with algebraic methods in constraint satisfaction problems.

Findings

New topological proof of the Hell-Nešetřil theorem

Clarifies the complexity classification of graph homomorphism problems

Bridges topological combinatorics with algebraic CSP approaches

Abstract

We provide a new proof of a theorem of Hell and Ne\v{s}et\v{r}il [J. Comb. Theory B, 48(1):92-110, 1990] using tools from topological combinatorics based on ideas of Lov\'asz [J. Comb. Theory, Ser. A, 25(3):319-324, 1978]. The Hell-Ne\v{s}et\v{r}il Theorem provides a dichotomy of the graph homomorphism problem. It states that deciding whether there is a graph homomorphism from a given graph to a fixed graph $H$ is in P if $H$ is bipartite (or contains a self-loop), and is NP-complete otherwise. In our proof we combine topological combinatorics with the algebraic approach to constraint satisfaction problem.