Toward Vu's conjecture
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914027020484608 |
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| author | Bradshaw, Peter Dhawan, Abhishek Methuku, Abhishek Wigal, Michael C. |
| author_facet | Bradshaw, Peter Dhawan, Abhishek Methuku, Abhishek Wigal, Michael C. |
| contents | In 2002, Vu conjectured that graphs of maximum degree $Δ$ and maximum codegree at most $ζΔ$ have chromatic number at most $(ζ+o(1))Δ$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the ``dense regime,'' when $ζ$ is close to $1$, by Hurley, de Verclos, and Kang.
In this paper we provide the first progress in the sparse regime $ζ\ll 1$, the case of primary interest to Vu. We show that there exists $ζ_0 > 0$ such that for all $ζ\in [\log^{-32}Δ,ζ_0]$, the following holds: if $G$ is a graph with maximum degree $Δ$ and maximum codegree at most $ζΔ$, then $χ(G) \leq (ζ^{1/32} + o(1))Δ$. We derive this from a more general result that assumes only that the common neighborhood of any $s$ vertices is bounded rather than the codegrees of pairs of vertices. Our more general result also extends to the list coloring setting, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16818 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toward Vu's conjecture Bradshaw, Peter Dhawan, Abhishek Methuku, Abhishek Wigal, Michael C. Combinatorics Discrete Mathematics In 2002, Vu conjectured that graphs of maximum degree $Δ$ and maximum codegree at most $ζΔ$ have chromatic number at most $(ζ+o(1))Δ$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the ``dense regime,'' when $ζ$ is close to $1$, by Hurley, de Verclos, and Kang. In this paper we provide the first progress in the sparse regime $ζ\ll 1$, the case of primary interest to Vu. We show that there exists $ζ_0 > 0$ such that for all $ζ\in [\log^{-32}Δ,ζ_0]$, the following holds: if $G$ is a graph with maximum degree $Δ$ and maximum codegree at most $ζΔ$, then $χ(G) \leq (ζ^{1/32} + o(1))Δ$. We derive this from a more general result that assumes only that the common neighborhood of any $s$ vertices is bounded rather than the codegrees of pairs of vertices. Our more general result also extends to the list coloring setting, which is of independent interest. |
| title | Toward Vu's conjecture |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2508.16818 |