On the Borodin--Kostochka conjecture for graphs with large maximum degree

Feng Liu; Shuang Sun; Yan Wang; Jiasheng Zeng

On the Borodin--Kostochka conjecture for graphs with large maximum degree

Combinatorics 2026-05-12 v3

Authors: Feng Liu , Shuang Sun , Yan Wang , Jiasheng Zeng

Abstract

The Borodin--Kostochka conjecture states that every graph $G$ with maximum degree $\Delta(G)\ge 9$ satisfies $\chi(G)\le \max\{\omega(G),\Delta(G)-1\}$ . In this paper, we verify this conjecture for graphs with sufficiently large maximum degree. More precisely, we prove that every graph $G$ with maximum degree $\Delta \ge 5.3\times 10^6$ and clique number $\omega(G)<\Delta$ satisfies $\chi(G)\le \Delta-1$ . This improves a longstanding result of Reed.

Keywords

hadwiger conjecture graph theory graph cycles

Cite

@article{arxiv.2603.16670,
  title  = {On the Borodin--Kostochka conjecture for graphs with large maximum degree},
  author = {Feng Liu and Shuang Sun and Yan Wang and Jiasheng Zeng},
  journal= {arXiv preprint arXiv:2603.16670},
  year   = {2026}
}

On the Borodin--Kostochka conjecture for graphs with large maximum degree

Abstract

Keywords

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