Reconfiguration of Independent Transversals

Given integers Δ ≥ and t ≥ 2 Δ, consider a graph of maximum degree (Formula presented.) and a partition of its vertices into blocks of size at least t. By a seminal result of Haxell, there is an independent set of the graph that is transversal to the blocks, a so-called independent transversal. We show that, if moreover t ≥ 2 Δ + 1, then every independent transversal can be transformed within the space of independent transversals to any other through a sequence of one-vertex modifications, showing connectivity of the so-called reconfigurability graph of independent transversals. This is sharp in that for t ≥ 2 Δ and Δ ≥ 2 the connectivity conclusion can fail. In this case, we show furthermore that in an essential sense it can only fail for the disjoint union of copies of the complete bipartite graph K_ΔΔ. This constitutes a qualitative refinement of Haxell's theorem.