A topological proof of the Hell-Ne\v{s}et\v{r}il dichotomy
Computational Complexity
2025-01-14 v2 Algebraic Topology
Combinatorics
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 is in P if 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.
Keywords
Cite
@article{arxiv.2409.12627,
title = {A topological proof of the Hell-Ne\v{s}et\v{r}il dichotomy},
author = {Sebastian Meyer and Jakub Opršal},
journal= {arXiv preprint arXiv:2409.12627},
year = {2025}
}
Comments
This version corrects a mistake in the proof of Theorem 3.2