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A topological proof of the Hell-Nešetřil dichotomy

Sebastian Meyer, Jakub Opršal

TL;DR

This work presents a new topological proof of the Hell–Nešetřil dichotomy for the $H$-colouring problem by blending Lovász-style topological combinatorics with the algebraic CSP framework centered on Taylor polymorphisms. The approach translates graph homomorphism questions into the topology of homomorphism complexes, showing that if $H$ is not bipartite and lacks a self-loop, then a sub-Taylor polymorphism forces contractibility and a fixed point, leading to NP-hardness; otherwise the problem is in P. The key technical contribution is proving that finite posets with sub-Taylor polymorphisms have contractible realizations, enabling a fixed-point argument that produces a self-loop in $H$. The results yield a conceptual, topology-based tractability criterion for finite-template CSPs and suggest broader generalizations to relational structures and promise CSPs, connecting homotopy theory with computational complexity in a new way.

Abstract

We provide a new proof of a theorem of Hell and Nešetřil [J. Comb. Theory B, 48(1):92-110, 1990] using tools from topological combinatorics based on ideas of Lovász [J. Comb. Theory, Ser. A, 25(3):319-324, 1978]. The Hell-Nešetřil 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.

A topological proof of the Hell-Nešetřil dichotomy

TL;DR

This work presents a new topological proof of the Hell–Nešetřil dichotomy for the -colouring problem by blending Lovász-style topological combinatorics with the algebraic CSP framework centered on Taylor polymorphisms. The approach translates graph homomorphism questions into the topology of homomorphism complexes, showing that if is not bipartite and lacks a self-loop, then a sub-Taylor polymorphism forces contractibility and a fixed point, leading to NP-hardness; otherwise the problem is in P. The key technical contribution is proving that finite posets with sub-Taylor polymorphisms have contractible realizations, enabling a fixed-point argument that produces a self-loop in . The results yield a conceptual, topology-based tractability criterion for finite-template CSPs and suggest broader generalizations to relational structures and promise CSPs, connecting homotopy theory with computational complexity in a new way.

Abstract

We provide a new proof of a theorem of Hell and Nešetřil [J. Comb. Theory B, 48(1):92-110, 1990] using tools from topological combinatorics based on ideas of Lovász [J. Comb. Theory, Ser. A, 25(3):319-324, 1978]. The Hell-Nešetřil 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.
Paper Structure (10 sections, 20 theorems, 7 equations, 1 figure)

This paper contains 10 sections, 20 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs and the H-colouring problem
  4. Posets, simplicial complexes, and their geometric realization
  5. Homotopy and a fixed-point theorem
  6. Multihomomorphisms and homomorphism complexes
  7. Taylor operations
  8. Taylor and contractibility of finite posets
  9. Proof of the Hell–Nešetřil theorem
  10. Conclusion

Key Result

Theorem 1.1

If $H$ is bipartite or contains a loop, then the $H$-coloring problem is in P. If $H$ is not bipartite and contains no loop then the $H$-coloring problem is NP-complete.

Figures (1)

  • Figure 1: The graph $K_3$ and the topological space $\mathop{\mathrm{\mathsf{Hom}}}\nolimits(K_2,K_3)$ which is homeomorphic to a circle. The multihomomorphisms are labelled and the action of $\phi$ on them is denoted by dashed lines.

Theorems & Definitions (25)

  • Theorem 1.1: Hell--Nešetřil
  • Definition 2.1: $H$-colouring
  • Lemma 2.2: Larose and Zádori LaroseZ97, based on Stong Strong66
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5: a corollary of the Lefschetz fixed-point theorem
  • Definition 2.6: the poset of multihomomorphisms
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 15 more