On the Borodin--Kostochka conjecture for graphs with large maximum degree
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866916000613531648 |
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| author | Liu, Feng Sun, Shuang Wang, Yan Zeng, Jiasheng |
| author_facet | Liu, Feng Sun, Shuang Wang, Yan Zeng, Jiasheng |
| contents | The Borodin--Kostochka conjecture states that every graph $G$ with maximum degree $Δ(G)\ge 9$ satisfies $χ(G)\le \max\{ω(G),Δ(G)-1\}$. 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 $Δ\ge 5.3\times 10^6$ and clique number $ω(G)<Δ$ satisfies $χ(G)\le Δ-1$. This improves a longstanding result of Reed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16670 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Borodin--Kostochka conjecture for graphs with large maximum degree Liu, Feng Sun, Shuang Wang, Yan Zeng, Jiasheng Combinatorics The Borodin--Kostochka conjecture states that every graph $G$ with maximum degree $Δ(G)\ge 9$ satisfies $χ(G)\le \max\{ω(G),Δ(G)-1\}$. 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 $Δ\ge 5.3\times 10^6$ and clique number $ω(G)<Δ$ satisfies $χ(G)\le Δ-1$. This improves a longstanding result of Reed. |
| title | On the Borodin--Kostochka conjecture for graphs with large maximum degree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.16670 |