On light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5

In 1940, in attempts to solve the Four Color Problem, Henry Lebesgue gave an approximate description of the neighborhoods of 5vertices in the class P5 of 3-polytopes with minimum degree 5. Given a 3-polytope P , by w(P ) (h(P )) we denote the minimum degreesum (minimum of the maximum degrees) of the neighborhoods of 5vertices in P . A 5∗-vertex is a 5-vertex adjacent to four 5-vertices. It is known that if a polytope P in P5 has a 5∗-vertex, then h(P ) can be arbitrarily large. For each P without vertices of degrees from 6 to 9 and 5∗-vertices in P5, it follows from Lebesgue’s Theorem that w(P ) ≤ 44 and h(P ) ≤ 14. In this paper, we prove that every such polytope P satisfies w(P ) ≤ 42 and h(P ) ≤ 12, where both bounds are tight.