All tight descriptions of major 3-paths in 3-polytopes without 3-vertices

Abstract

A 3-path uvw is an (i,j,k)-path if $d(u)\le i$, $d(v)\le j$, and $d(w)\le k$, where d(x) is the degree of a vertex x. It is well-known that each 3-polytope has a vertex of degree at most 5, called minor. A description of 3-paths in a 3-polytope is minor or major if the central item of each its triplet is at most 5 or at least 6, respectively. Back in 1922, Franklin proved that each 3-polytope with minimum degree 5 has a (6,5,6)-path, which description is tight. Recently, Borodin and Ivanova extended Franklin's theorem by producing all the ten tight minor descriptions of 3-paths in the class ${\bf P_4}$ of 3-polytopes with minimum degree at least 4. In 2016, Borodin and Ivanova proved that each polytope with minimum degree 5 has a (5,6,6)-path, and there exists no tight description of 3-paths in this class of 3-polytopes other than {(6,5,6)} and {(5,6,6)}. The purpose of this paper is to prove that there exist precisely the following four major tight descriptions of 3-paths in ${\bf P_4}$:{(4,9,4),(4,7,5), (5,6,6)}, {(4,9,4),(5,7,6)}, {(4,9,5),(5,6,6)}, and {(5,9,6)}.