Characterizing crystallizations among Lins-Mandel 4-coloured graphs

In [Discr. Math. 57 (1985), 261-284], Lins and Mandel introduce a class of 3-manifolds represented by 4-coloured graphs S(b,l,t,c) depending on a 4-tuple (b,l,t,c) of positive integers; moreover, they prove that, if the following conditions hold $(b,c)=1$, $(l,t)=1$, $c=(-1)^t$ if l odd, then S(b,l,t,c) is a crystallization of an (orientable) 3-manifold.In this paper we show that the above conditions are also necessary: hence, they characterize crystallizations among Lins-Mandel graphs.