On Vizing's problem for triangle-free graphs

Ross J. Kang; Matthieu Rosenfeld

On Vizing's problem for triangle-free graphs

Combinatorics 2025-09-05 v4

Authors: Ross J. Kang , Matthieu Rosenfeld

Abstract

We prove that $\chi(G) \le \lceil (\Delta+1)/2\rceil+1$ for any triangle-free graph $G$ of maximum degree $\Delta$ provided $\Delta \ge 524$ . This gives tangible progress towards an old problem of Vizing, in a form cast by Reed. We use a method of Hurley and Pirot, which in turn relies on a new counting argument of the second author.

Keywords

graph theory

Cite

@article{arxiv.2309.10876,
  title  = {On Vizing's problem for triangle-free graphs},
  author = {Ross J. Kang and Matthieu Rosenfeld},
  journal= {arXiv preprint arXiv:2309.10876},
  year   = {2025}
}

Comments

11 pages; we learned from Z Dvorak and R Wdowinski that a conjecture we posed was known, so Sec 3 in v3 is adapted accordingly, and the suggestions of referees are also incorporated

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