The average edge order of 3-manifold coloured triangulations

If K is a triangulation of a closed 3-manifold M with $E_0(K)$ edges and $F_0(K)$ triangles, then the average edge order of K is defined to be $\mu_0(K) = 3F_0(K)/E_0(K) In [8], the relations between this quantity and the topology of M are investigated, especially in the case of $\mu_0(K)$ being small (where the study relies on Oda's classification of triangulations of $S^2$ up to eight vertices. In the present paper, the attention is fixed upon the average edge order of coloured triangulations; surprisingly enough, the obtained results are perfectly analogous to Luo-Stong' ones, and may be proved with little effort by means of edge-coloured graphs representing manifolds.