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Hall's theorem for reconfigurations and higher dimensional topological connectedness

Ronen Wdowinski

TL;DR

This work develops a unified topological framework to study reconfigurations of combinatorial structures tied to colorings and color classes. By extending Hall’s transversal theorem to reconfigurations, the authors prove both homotopical and homological versions of a topological Hall theorem, and generalize to higher dimensional colorful complexes and matroid settings. They derive connectivity results for reconfiguration graphs across independent transversals, rainbow matchings, bipartite hypergraph matchings, and list colorings, and apply these to discrete geometry through reconfigurations of colorful Helly, Carathéodory, and Tverberg theorems, resolving several conjectures. The results provide structural guarantees for solution spaces and offer new proofs and tighter conditions, though algorithmic diameter and mixing-time questions remain open for future work.

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 Carathéodory, and Tverberg theorems from discrete geometry, confirming a conjecture of Oliveros, Roldán, Soberón, and Torres.

Hall's theorem for reconfigurations and higher dimensional topological connectedness

TL;DR

This work develops a unified topological framework to study reconfigurations of combinatorial structures tied to colorings and color classes. By extending Hall’s transversal theorem to reconfigurations, the authors prove both homotopical and homological versions of a topological Hall theorem, and generalize to higher dimensional colorful complexes and matroid settings. They derive connectivity results for reconfiguration graphs across independent transversals, rainbow matchings, bipartite hypergraph matchings, and list colorings, and apply these to discrete geometry through reconfigurations of colorful Helly, Carathéodory, and Tverberg theorems, resolving several conjectures. The results provide structural guarantees for solution spaces and offer new proofs and tighter conditions, though algorithmic diameter and mixing-time questions remain open for future work.

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 Carathéodory, and Tverberg theorems from discrete geometry, confirming a conjecture of Oliveros, Roldán, Soberón, and Torres.

Paper Structure

This paper contains 25 sections, 58 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main theorems and topological tools
  3. Main theorems
  4. Further topological notions
  5. Some complexes and their topological connectedness
  6. Proof of topological Hall theorem for reconfigurations
  7. A Sperner-type lemma
  8. Homotopical connectedness
  9. Homological connectedness
  10. Higher dimensional topological Hall theorems
  11. Proof of Theorem \ref{['thm:topological-reconfiguration-complex']}
  12. A nerve variation
  13. Intersections of complexes and matroids
  14. Reconfiguration results
  15. Higher dimensional connectedness
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $G$ be a graph with maximum degree $\Delta$, and let $\mathcal{V} = \{V_1, \ldots, V_n\}$ be a partition of $V(G)$ such that $|V_i| \ge 2\Delta$ for all $i$. If $G[V_I]$ is not the disjoint union of $|I|$ copies of the complete bipartite graph $K_{\Delta,\Delta}$, for all nonempty $I \subseteq [

Figures (2)

  • Figure 1: An illustration of our Sperner-type lemma (Lemma \ref{['lem:Sperner']}) on the prism $P^2 = \Delta^1 \times [0,1]$. The associated graph $G$ is represented by dashed segments which connect adjacent colorful simplices. There is an odd number of paths that connect a colorful simplex in $\Delta^1 \times \{0\}$ to a colorful simplex in $\Delta^1 \times \{1\}$.
  • Figure 2: An illustration of the colorful complex $\mathbf{Col}(\mathcal{C}, \mathcal{V})$ and the corresponding polyhedral complex $\mathbf{Col}^{\mathrm{P}}(\mathcal{C}, \mathcal{V})$, where $\mathcal{C}$ is the simplex with vertices $\{u_0, v_0, u_1, v_1\}$, and $\mathcal{V} = \{V_0, V_1\}$ is given by $V_0 = \{u_0, v_0\}$, $V_1 = \{u_1, v_1\}$.

Theorems & Definitions (83)

  • Theorem 1.1: buys2025reconfiguration
  • Theorem 1.2
  • Theorem 2.1: meshulam2003domination
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4: szabo2006extremal
  • Proposition 2.5: aharoni2006intersectionaharoni2015cooperative
  • Theorem 2.6: aharoni2006intersectionaharoni2000hallmeshulam2001clique
  • Theorem 2.7: aharoni2005eigenvaluesaharoni2000hall
  • Proposition 2.8: bjorner1992homology
  • ...and 73 more