Cut-homotopies and the complexity of edge-coloring problems

Alexey Barsukov; Roman Feller; Maximilian Hadek; Davide Perinti

Cut-homotopies and the complexity of edge-coloring problems

Combinatorics 2026-07-10 v1 Computational Complexity

Authors: Alexey Barsukov , Roman Feller , Maximilian Hadek , Davide Perinti

Abstract

We study the computational complexity of problems that ask if a given graph admits an edge-coloring that does not contain an edge-colored clique from some fixed finite family. We show that every such problem is poly-time equivalent to a Constraint Satisfaction Problem, yielding a P vs. NP-complete dichotomy. Our main contribution lies in the reduction from the CSP to the coloring problem where we apply methods from Ramsey theory and a novel notion of cut-homotopy.

Keywords

induced erdos-posa property planar minors maximum independent set quasi-polynomial algorithm tree-width graph theory

Cite

@article{arxiv.2607.09631,
  title  = {Cut-homotopies and the complexity of edge-coloring problems},
  author = {Alexey Barsukov and Roman Feller and Maximilian Hadek and Davide Perinti},
  journal= {arXiv preprint arXiv:2607.09631},
  year   = {2026}
}