2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58850As previously known, all 3-manifolds of genus two can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an ``elementary transformation'' on these 6-tuples which changes the associated graph but does not change the represented manifold. This operation is a useful tool in the classification problem for 3-manifolds of genus two; in fact, it allows to define an equivalence relation on ``admissible'' 6-tuples so that equivalent 6-tuples represent the same manifold. Different equivalence classes can represent the same manifold; however, equivalence classes ``almost always'' contain infinitely many 6-tuples. Finally, minimal representatives of the equivalence classes are described.27 pages, 10 figures, to appear on Journal of Combinatorial Theory Series BGeometric Topology57M12, 57M15 (Primary); 57M25, 05C10 (Secondary)text