Independent transversals in bipartite correspondence-covers

TL;DR

This paper proves that under certain size and degree conditions, bipartite graphs with a specific correspondence structure contain an independent transversal, advancing understanding of graph matchings and transversals.

Contribution

The authors establish near-optimal size bounds for independent transversals in bipartite graphs with correspondence covers, including asymmetric variants.

Findings

Bound on part sizes is asymptotically sharp up to a factor of 2.

Existence of independent transversals under specified degree and size conditions.

Results hold for sufficiently large maximum degree D.

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.