Abstract :
It is shown that for every sequence of nonnegative integers (p3, P5, P6, …, Pn) satisfying the equation ∑k⩾3(4-k) Pk=8, which follows from the well-known Eulerʹs formula, there exists an integer p4 and a planar 4-valent 3-connected graph G that has exactly pkk-gonal faces for all 3⩽k⩽n and pk=0, otherwise, and that is cut-through Eulerian. This is an extension of Grünbaumʹs theorem.
