Defect-Path Routing in Factor-Critical Graphs

Abstract. We prove that in every factor-critical graph, any two vertex-disjoint compatible edges lie on a common defect path. Specifically, if F is a factor-critical graph and e, f are vertex-disjoint edges that are simultaneously contained in some near-perfect matching of F , then there exist vertices p ̸= q and matchings K ∈PM(F−p), L ∈PM(F−q) such that the unique p–q path in the symmetric difference K△L contains both e and f . The proof is short and rests on a single structural observation: a forbidden edge cannot lie on a cycle component of a symmetric difference, since a cycle flip would produce a perfect matching containing that edge, contradicting its forbidden status. As an application, we show that in any brick H, every pair of compatible edges not incident to a common vertex lies on a conformal even cycle.