Hall's theorem for reconfigurations and higher dimensional topological connectedness

TL;DR

This paper extends topological Hall's theorem to higher dimensions, providing conditions for reconfiguring colorful simplices through vertex swaps while maintaining certain topological and combinatorial properties.

Contribution

It introduces a topological Hall theorem for homological connectedness of colorful simplices and generalizes it to matroids, with applications to reconfiguration problems.

Findings

Connectedness conditions for reconfiguration graphs in various combinatorial structures

Alternative proof for maximum degree condition in independent transversal reconfigurability

Tight reconfiguration versions of geometric theorems like Helly, Carathéodory, and Tverberg

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.