Asymmetric list sizes in bipartite graphs
Noga Alon, Stijn Cambie, Ross J. Kang
TL;DR
This paper investigates asymmetric list-colouring in bipartite graphs, focusing on (k_A,k_B)-choosability for graphs with partite sets A,B of maximum degrees Δ_A, Δ_B. It develops two main probabilistic frameworks to obtain sufficient conditions: independent transversals via the Lovász Local Lemma and coupon-collector analyses, both leading to concrete bounds ensuring (k_A,k_B)-choosability. It then connects the complete bipartite case to a design-theoretic extremal parameter overline{M}(k_1,k_2,ℓ) 007F and derives necessary conditions, including asymptotic behavior for ch(K_{n,n}) and related thresholds. The work shows complete bipartite graphs are not always extremal, provides sharp boundary-case results, and establishes a degree-based transfer principle for non-choosability, thereby advancing asymmetric generalizations of longstanding conjectures in list-colouring and linking to combinatorial design theory. Overall, the results lay groundwork toward near-optimal asymmetric thresholds and stimulate further exploration of the interaction between list-colouring, transversals, coupon collection, and Steiner-system-inspired extremal parameters.
Abstract
Given a bipartite graph with parts $A$ and $B$ having maximum degrees at most $Δ_A$ and $Δ_B$, respectively, consider a list assignment such that every vertex in $A$ or $B$ is given a list of colours of size $k_A$ or $k_B$, respectively. We prove some general sufficient conditions in terms of $Δ_A$, $Δ_B$, $k_A$, $k_B$ to be guaranteed a proper colouring such that each vertex is coloured using only a colour from its list. These are asymptotically nearly sharp in the very asymmetric cases. We establish one sufficient condition in particular, where $Δ_A=Δ_B=Δ$, $k_A=\log Δ$ and $k_B=(1+o(1))Δ/\logΔ$ as $Δ\to\infty$. This amounts to partial progress towards a conjecture from 1998 of Krivelevich and the first author. We also derive some necessary conditions through an intriguing connection between the complete case and the extremal size of approximate Steiner systems. We show that for complete bipartite graphs these conditions are asymptotically nearly sharp in a large part of the parameter space. This has provoked the following. In the setup above, we conjecture that a proper list colouring is always guaranteed * if $k_A \ge Δ_A^\varepsilon$ and $k_B \ge Δ_B^\varepsilon$ for any $\varepsilon>0$ provided $Δ_A$ and $Δ_B$ are large enough; * if $k_A \ge C \logΔ_B$ and $k_B \ge C \logΔ_A$ for some absolute constant $C>1$; or * if $Δ_A=Δ_B = Δ$ and $ k_B \ge C (Δ/\logΔ)^{1/k_A}\log Δ$ for some absolute constant $C>0$. These are asymmetric generalisations of the above-mentioned conjecture of Krivelevich and the first author, and if true are close to best possible. Our general sufficient conditions provide partial progress towards these conjectures.