Tight constructions for reconfigurations of independent transversals

TL;DR

This paper characterizes the structure of certain exceptional cases in reconfiguration graphs of independent transversals, revealing a simple construction behind complex instances.

Contribution

It provides an exact characterization of the partition structures in the exceptional cases of a reconfiguration connectivity theorem.

Findings

Identifies the structure of exceptional instances where the reconfiguration graph is disconnected.

Shows that these instances are generated by a simple constructive procedure.

Reveals a rich variety of such exceptional configurations.

Abstract

For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a reconfiguration graph on independent transversals of $(G,\mathcal{U})$ is connected, meaning any independent transversal can be transformed into any other one through a sequence of one-vertex modifications while always maintaining an independent transversal. Analogous to a theorem of Haxell, they proved that this is the case if $G$ has maximum degree $\Delta$ and each block of $\mathcal{U}$ has size at least $2\Delta$, except if the union of some $k \ge 1$ blocks of $\mathcal{U}$ induces $k$ disjoint copies of the complete bipartite graph $K_{\Delta, \Delta}$ in $G$. Solving one of their problems, we exactly characterize the partition structure in the latter exceptional instances of their theorem, showing that there is a rich variety of them but they are generated by a simple constructive procedure.