A precise condition for independent transversals in bipartite covers

TL;DR

This paper establishes a sharp, precise condition involving degrees and partition sizes that guarantees the existence of an independent transversal in bipartite graphs, extending classical results in the field.

Contribution

The paper introduces a new sharp condition for independent transversals in bipartite graphs, refining previous results and providing a more precise criterion.

Findings

Proves that D_A/k_B + D_B/k_A ≤ 1 guarantees an independent transversal.

Shows the condition is sharp and cannot be improved.

Extends classical results on independent transversals to bipartite graphs.

Abstract

Given a bipartite graph $H=(V=V_A\cup V_B,E)$ in which any vertex in $V_A$ (resp.~$V_B$) has degree at most $D_A$ (resp.~$D_B$), suppose there is a partition of $V$ that is a refinement of the bipartition $V_A\cup V_B$ such that the parts in $V_A$ (resp.~$V_B$) have size at least $k_A$ (resp.~$k_B$). We prove that the condition $D_A/k_B+D_B/k_A\le 1$ is sufficient for the existence of an independent set of vertices of $H$ that is simultaneously transversal to the partition, and show moreover that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author and the other due to Szab\'o and Tardos.