Estimating Matveev's complexity via crystallization theory

In [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology Appl. 144(1–3) (2004) 201–209], a graph-theoretical approach to Matveev's complexity   computation is introduced, yielding the complete classification of closed non-orientable 3-manifolds up to complexity six. The present paper follows the same point-of view, making use of crystallization theory and related results (see [M. Ferri, Crystallisations of 2-fold branched coverings of S3S3, Proc. Amer. Math. Soc. 73 (1979) 271–276; M.R. Casali, Coloured knots and coloured graphs representing 3-fold simple coverings of S3S3, Discrete Math. 137 (1995) 87–98; M.R. Casali, From framed links to crystallizations of bounded 4-manifolds, J. Knot Theory Ramifications 9(4) (2000) 443–458]) in order to significantly improve existing estimations for complexity of both 2-fold and three-fold simple branched coverings (see [O.M. Davydov, The complexity of 2-fold branched coverings of a 3-sphere, Acta Appl. Math. 75 (2003) 51–54] and [O.M. Davydov, Estimating complexity of 3-manifolds as of branched coverings, talk-abstract, Second Russian-German Geometry Meeting dedicated to 90-anniversary of A.D.Alexandrov, Saint-Petersburg, Russia, June 2002]) and 3-manifolds seen as Dehn surgery (see [G. Amendola, An algorithm producing a standard spine of a 3-manifold presented by surgery along a link, Rend. Circ. Mat. Palermo 51 (2002) 179–198]).