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

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 $\mathbf{P_4}$ 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 $\mathbf{ P_4}$: $\{(4,9,4),(4,7,5),(5,6,6)\}$, $\{(4,9,4),(5,7,6)\}$, $\{(4,9,5),(5,6,6)\}$, and $\{(5,9,6)\}$.