Universal bounded torsion for matching complexes without genus restriction

Determine whether torsion in the integral homology groups of matching complexes M(G) is universally bounded over all finite undirected graphs G without any genus constraint; specifically, for each homological degree i ≥ 0, ascertain the existence of an integer m(i) that annihilates the torsion subgroup of H_i(M(G); Z) for every finite graph G.

Background

Miyata and Ramos proved that for undirected graphs with bounded genus, the homology of their matching complexes has universally bounded torsion. This paper extends categorical minor methods to quivers and discusses consequences for related complexes.

The authors explicitly note that extending the bounded-torsion result to all graphs (without a genus bound) is conjectural. Resolving this would generalize powerful Noetherian and bounded-torsion phenomena from bounded-genus settings to the full class of finite graphs, impacting the paper of torsion in matching complexes.