Optimal binding function for (cap,even hole)-free graphs with no short odd holes

Chenglong Deng; Xuding zhu

Optimal binding function for (cap,even hole)-free graphs with no short odd holes

Combinatorics 2026-07-30 v1

Authors: Chenglong Deng , Xuding zhu

Abstract

A hole in a graph is an induced cycle of length at least $4$ . A cap is a hole together with a vertex adjacent to exactly two consecutive vertices of it. Chen, Xu and Xu conjectured that if $q\ge2$ and $G$ is a $(\mathrm{cap},\mathrm{even\ hole})$ -free graph with no odd hole of length at most $2q-1$ , then $\chi(G)\le \left\lceil \frac{2q+1}{2q}\omega(G)\right\rceil.$ They confirmed the conjecture for $q \le 3$ . In this paper, we prove the conjecture for all $q \ge 3$ . As a corollary, we prove that for such a graph $G$ , $\chi_f(G)\le \frac{2q+1}{2q}\omega(G).$

Cite

@article{arxiv.2607.27850,
  title  = {Optimal binding function for (cap,even hole)-free graphs with no short odd holes},
  author = {Chenglong Deng and Xuding zhu},
  journal= {arXiv preprint arXiv:2607.27850},
  year   = {2026}
}