Hall's theorem for reconfigurations and higher dimensional topological connectedness

Abstract

One widely applied sufficient condition for the existence of a colorful simplex in a vertex-colored simplicial complex is a topological extension of Hall's transversal theorem due to Aharoni, Haxell, and Meshulam. We prove a similar topological Hall theorem that provides a sufficient condition for being able to transform any colorful simplex into any other through a sequence of one-vertex swaps while always maintaining a colorful simplex, meaning that the associated reconfiguration graph is connected. In fact, we prove a generalized topological Hall theorem about the homological connectedness of the space of colorful simplices, as well as a matroidal generalization of this result. We deduce sufficient conditions for reconfiguration graphs to be connected for various combinatorial structures of interest such as independent transversals in graphs, matchings in bipartite hypergraphs, and intersections of matroids. In particular, we give an alternative proof of a maximum degree condition for independent transversal reconfigurability by Buys, Kang, and Ozeki. We also deduce tight reconfiguration versions of the colorful Helly, colorful Carath\'eodory, and Tverberg theorems from discrete geometry, confirming a conjecture of Oliveros, Rold\'an, Sober\'on, and Torres.