Reconfiguration of Independent Transversals

TL;DR

We address the connectivity of the space of independent transversals in graphs of maximum degree $Δ$ under a $t$-thick partition. By developing a reconfiguration framework and employing augmenting sequences alongside a descent argument in the style of Haxell, we show that if $t\ge 2Δ+1$ the reconfigurability graph on independent transversals is connected, so any two transversals can be joined by single-vertex moves while maintaining independence. The bound is tight: for $t=2Δ$ connectivity can fail, with the obstruction essentially the disjoint union of $|\mathcal U|$ copies of $K_{Δ,Δ}$. This provides a reconfiguration-analogue of Haxell's theorem and offers a foundation for studying dynamics and structure of transversal spaces in extremal combinatorics, with potential implications for Markov chains and coloring reconfiguration problems.

Abstract

Given integers $Δ\ge 2$ and $t\ge 2Δ$, suppose there is a graph of maximum degree $Δ$ and a partition of its vertices into blocks of size at least $t$. By a seminal result of Haxell, there must be some independent set of the graph that is transversal to the blocks, a so-called independent transversal. We show that, if moreover $t\ge2Δ+1$, 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 $t=2Δ$ (and $Δ\ge 2$) 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 $K_{Δ,Δ}$. This constitutes a qualitative strengthening of Haxell's theorem.

Figures (1)

• Figure 1: An example of an instance $\mathcal{G}'$ obtained by swapping $B$ and $U\setminus A$ in $\mathcal{G}$. The sets $S_1 = \{u_0,u_1,u_2,u_3\}$ and $S_2 = \{v_0,v_1,y_2,y_3\}$ are two examples of independent transversals in $\mathcal{G}'$ an element from both $M(S_1)$ and $M(S_2)$ is drawn in $\mathcal{G}$.

Theorems & Definitions (20)

• Theorem 1.1: Haxell1995Haxell2001
• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Haxell's theorem
• Lemma 2.3
• proof
• Corollary 2.4