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

TL;DR

This paper improves the upper bound on the list chromatic number of bipartite graphs, showing it is less than a certain fraction of rac{ ext{max degree}}{ ext{log max degree}}, advancing toward a longstanding conjecture.

Contribution

It establishes a new upper bound for bipartite graph choosability, demonstrating it is strictly less than the known bounds for triangle-free graphs, thus highlighting a fundamental difference.

Findings

For large elta, elta/ log elta bounds the list chromatic number.

The bound is improved to less than (rac;4/5 - psilon) elta/ log elta.

Supports the conjecture that bipartite graphs have lower list chromatic numbers than general triangle-free graphs.

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.