6-Valent analogues of Eberhard's theorem

It is shown that for every sequence of non-negative integers (p _n|1≦n≠3) satisfying the equation {ie19-1} (respectively, =0) there exists a 6-valent, planar (toroidal, respectively) multi-graph that has precisely p _n n gonal faces for all n, 1≦n≠3. This extends Eberhard's theorem that deals, in a similar fashion, with 3-valent, 3-connected planar graphs; the equation involved follows from the famous Euler's equation.