2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/218464Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.The paper is totally rewritten. The title has been changed. 9 pages, 5 figuresGeometric Topology57N10text