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arXiv · 2409.01513

Bipartite graphs are $(\frac{4}{5}-\varepsilon) \frac{\Delta}{\log \Delta}$-choosable

Peter Bradshaw·Bojan Mohar·Ladislav Stacho
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Abstract

Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $\Delta$, then the choosability (or list chromatic number) of $G$ satisfies $\chi_{\ell}(G) = O \left ( \log \Delta \right )$. Currently, the best known upper bound for $\chi_{\ell}(G)$ is $(1 + o(1)) \frac{\Delta}{\log \Delta}$, 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 $\Delta$ satisfies $\chi_{\ell}(G) < (\frac{4}{5} -\varepsilon) \frac{\Delta}{\log \Delta}$. 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.

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Peter Bradshaw, Bojan Mohar, Ladislav Stacho. 2024-09-03. Bipartite graphs are $(\frac{4}{5}-\varepsilon) \frac{\Delta}{\log \Delta}$-choosable. https://arxiv.org/abs/2409.01513

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