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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.01513 |
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| _version_ | 1866916379687387136 |
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| author | Bradshaw, Peter Mohar, Bojan Stacho, Ladislav |
| author_facet | Bradshaw, Peter Mohar, Bojan Stacho, Ladislav |
| contents | Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $Δ$, then the choosability (or list chromatic number) of $G$ satisfies $χ_{\ell}(G) = O \left ( \log Δ\right )$. Currently, the best known upper bound for $χ_{\ell}(G)$ is $(1 + o(1)) \fracΔ{\log Δ}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $Δ$ satisfies $χ_{\ell}(G) < (\frac{4}{5} -\varepsilon) \fracΔ{\log Δ}$. This improved upper bound suggests that list coloring is fundamentally different for bipartite graphs than for triangle-free graphs and hence gives a step toward solving the conjecture of Alon and Krivelevich. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01513 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bipartite graphs are $(\frac{4}{5}-\varepsilon) \fracΔ{\log Δ}$-choosable Bradshaw, Peter Mohar, Bojan Stacho, Ladislav Combinatorics 05C15 Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $Δ$, then the choosability (or list chromatic number) of $G$ satisfies $χ_{\ell}(G) = O \left ( \log Δ\right )$. Currently, the best known upper bound for $χ_{\ell}(G)$ is $(1 + o(1)) \fracΔ{\log Δ}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $Δ$ satisfies $χ_{\ell}(G) < (\frac{4}{5} -\varepsilon) \fracΔ{\log Δ}$. This improved upper bound suggests that list coloring is fundamentally different for bipartite graphs than for triangle-free graphs and hence gives a step toward solving the conjecture of Alon and Krivelevich. |
| title | Bipartite graphs are $(\frac{4}{5}-\varepsilon) \fracΔ{\log Δ}$-choosable |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2409.01513 |