Constructing certain families of 3‐polytopal graphs

Let n ≥ 3 $n\ge 3$ and R ${R}_{n}$ be a 3-polytopal graph such that for every ≤ i $3\le i\le n$ , has at least one vertex of degree $i$ . We find the minimal count then describe an algorithm to construct graphs A dual statement may formulated faces 3-polytopes. The ideas behind generalise readily solve related problems. Moreover, given 3-polytope T l ${T}_{l}$ comprising all l$ $l$ fixed, we define output > $n\gt ${T}_{n}$ initial is subgraph asymptotically optimal, in sense it matches aforementioned up order magnitude, as $n$ gets large. In fact, only lose small quantity on coefficient second highest term, this taken please, with tradeoff first constructing accordingly large auxiliary graph.