The present paper looks at Matveev's complexity (introduced in 1990 and based on the existence of a simple spine for each compact 3-manifold: see [Acta Appl. Math. 19 (1990), 101-130]) through another combinatorial theory for representing 3-manifolds, which makes use of particular edge-coloured graphs, called crystallizations. Crystallization catalogue $\tilde C^{26}$ for closed non-orientable 3-manifolds (due to [Acta Appl. Math. 54 (1999), 75-97]) is proved to yield upper bounds for Matveev's complexity of any involved 3-manifold. As a consequence, an improvement of Amendola and Martelli classification of closed non-orientable irreducible and $P^2$-irreducible 3-manifolds up to complexity c=6 is obtained.

Computing Matveev’s complexity of non-orientable 3-manifolds via crystallization theory / Casali, Maria Rita. - In: TOPOLOGY AND ITS APPLICATIONS. - ISSN 0166-8641. - STAMPA. - 144:(1-3)(2004), pp. 201-209. [10.1016/j.topol.2004.04.010]

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