Independent transversals in bipartite correspondence-covers
Stijn Cambie, Ross J. Kang
TL;DR
This work studies independent transversals in bipartite correspondence-covers, extending the classical list-colouring framework to covers where edges between L(v) and L(v') form a matching for every G-edge. The authors develop a probabilistic approach, employing a hypergraph transversal framework and the Lovász Local Lemma to obtain existence results under regimes where each part has size |L(v)| that scales like (1+ε)D/ log D with D the maximum degree of H; they prove a near-optimal bound (tight up to a factor of 2) and derive several asymmetric variants. Additionally, they establish an asymmetric analogue of Haxell's theorem, enabling independent transversals by first handling A_H via an auxiliary graph with controlled degree and then extending to B_H under a two-element list constraint. The results advance the understanding of when independent transversals exist in correspondence-covers, connecting to adjacency-structure constraints and generating near-sharp thresholds that align with, and extend, prior work on list-colouring and correspondence-colouring of bipartite graphs.
Abstract
Suppose $G$ and $H$ are bipartite graphs and $L: V(G)\to 2^{V(H)}$ induces a partition of $V(H)$ such that the subgraph of $H$ induced between $L(v)$ and $L(v')$ is a matching whenever $vv'\in E(G)$. We show for each $\varepsilon>0$ that, if $H$ has maximum degree $D$ and $|L(v)| \ge (1+\varepsilon)D/\log D$ for all $v\in V(G)$, then $H$ admits an independent transversal with respect to $L$, provided $D$ is sufficiently large. This bound on the part sizes is asymptotically sharp up to a factor $2$. We also show some asymmetric variants of this result.