A structural property of convex 3-polytopes (Q1376488)

A \((d_1,d_2, \dots, d_k)\)-path on a convex 3-polytope (3-connected planar graph) is one whose successive vertices have degrees \(d_1\), \(d_2, \dots, d_k\). The author shows here that a 3-polytope always has \((a,b,c)\)-paths for certain restrictions on the degrees. In particular, generalizing the (best possible) cases \(k=1\) \((a\leq 5)\) and \(k=2(a+b\leq 13\), due to \textit{A. Kotzig} [Mat.-Fyz. Čas., Slovensk. Akad. Vied 5, 101-103 (1955)]), he shows that \(a+b +c\leq 23\) is attainable; he conjectures that 21 is the correct bound.