Low and light $5$-stars in $3$-polytopes with minimum degree $5$ and restrictions on the degrees of major vertices

In 1940, in attempts to solve the Four Color Problem, Henry Lebesgue gave an approximate description of the neighborhoods of $5$-vertices in the class $\mathbf P_5$ of 3-polytopes with minimum degree $5$. This description depends on $32$ main parameters. Very few precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in $\mathbf P_5$. Given a $3$-polytope $P$, denote the minimum of the maximum degrees (height) of the neighborhoods of $5$-vertices (minor $5$-stars) in $P$ by $h(P)$. Jendrol'and Madaras in 1996 showed that if a polytope $P$ in $\mathbf P_5$ is allowed to have a $5$-vertex adjacent to four $5$-vertices (called a minor $(5,5,5,5,\infty)$-star), then $h(P)$ can be arbitrarily large. For each $P^*$ in $\mathbf P_5$ with neither vertices of the degree from $6$ to $8$ nor minor $(5,5,5,5,\infty)$-star, it follows from Lebesgue's Theorem that $h(P^*)\le17$. We prove in particular that every such polytope $P^*$ satisfies $h(P^*)\le12$, and this bound is sharp. This result is best possible in the sense that if vertices of one of degrees in $\{6,7,8\}$ are allowed but those of the other two forbidden, then the height of minor $5$-stars in $\mathbf P_5$ under the absence of minor $(5,5,5,5,\infty)$-stars can reach $15$, $17$, or $14$, respectively.