A Planar Counterexample to the Sphere-Wedge Conjecture for Matching Complexes

Abstract

The matching complex of a finite simple graph consists of its sets of pairwise disjoint edges. We prove that the matching complex of the planar pentagonal bipyramid simplicially collapses to a triangulated torus. This disproves the conjecture that matching complexes of planar graphs are contractible or homotopy equivalent to wedges of spheres. The counterexample has the smallest possible number of graph vertices and is unique among seven-vertex planar graphs. More generally, let D_m = C_m * K̅₂ be the graph join of a cycle with two nonadjacent apices. We determine the homotopy type of its matching complex for every m ≥ 3: the only exception to a wedge of explicitly counted spheres is the torus at m = 5. On each original complex we construct an acyclic Morse matching that simultaneously minimizes every critical-cell count and their total. We also treat every nonempty retained spoke set at one apex while keeping the other apex full. When m ≡ 2 (mod 3), both spoke sets may be arbitrary, with all rim edges retained. An auxiliary bipartite graph determines the natural integral homology, and its spanning forests give pairings on actual critical faces. The proof combines elementary collapses, an integral incidence calculation, and opposite symmetry actions that force the only possible higher differential to vanish.