Given integers $\Delta\ge 2$ and $t\ge 2\Delta$, suppose there is a graph of maximum degree $\Delta$ 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 $t\ge2\Delta+1$, 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 $t=2\Delta$ (and $\Delta\ge 2$) 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 $K_{\Delta,\Delta}$. This constitutes a qualitative strengthening of Haxell's theorem.

翻译：给定整数 $\Delta\ge 2$ 和 $t\ge 2\Delta$，假设存在一个最大度为 $\Delta$ 的图，其顶点被划分为大小至少为 $t$ 的块。根据 Haxell 的一个开创性结果，该图必然存在一个独立集，该独立集与所有块横截，即所谓的独立横截集。我们证明，如果进一步满足 $t\ge2\Delta+1$，则每一个独立横截集都可以在独立横截集的空间内，通过一系列单顶点修改，变换为任何其他独立横截集，从而证明了独立横截集的可重构性图是连通的。这一结论是尖锐的，因为对于 $t=2\Delta$（且 $\Delta\ge 2$），连通性结论可能不成立。在这种情况下，我们进一步证明，从本质上讲，连通性失败的情况只能出现在完全二分图 $K_{\Delta,\Delta}$ 的不相交并集上。这构成了对 Haxell 定理的一种定性强化。