On face-vectors of 5-valent convex 3-polytopes (Q1331274)

A convex 3-dimensional polytope is \(s\)-valent if all its vertices are \(s\)-valent. For a convex 3-polytope \(P\), let \(p_ k(P)\) denote the number of its \(k\)-gonal faces. A sequence \((p_ k(P)\mid k\geq 3)\) associated with \(P\) is called the face-vector of \(P\). A sequence \(q= (q_ k\mid k\geq 3)\) of nonnegative integers is 5-realizable if there exists a 5-valent 3-polytope \(P\) such that \(p_ k(P)= q_ k\) for all \(k\geq 3\). This paper describes a new necessary condition and a few new sufficient conditions on the 5-realizability of sequences.