Upper and lower bounds of the valence-functional

For the non-negative integer g let (M, g) denote the closed orientable 2-dimensional manifold of genus g. K-realizations P of (M, g) are geometric cell-complexes in P with convex facets such that set (P) is homeomorphic to M. For K-realizations P of (M, g) and vertices v of P, val (v, P) denotes the number of edges of P incident with v and the weighted vertex-number Σ(val(v, P)-3) taken over all vertices of P is called valence-value v (P) of P. The valence-functional V, which is important for the determination of all possible f-vectors of K-realisations of (M, g), in connection with Eberhard's problem etc., is defined by V(g):=min[v(P)|P is a K-realization of (M,g)]. The aim of the note is to prove the inequality 2 g+1≦V(g)≦3 g+3 for every positive integer g.