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 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 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 P 5 is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5,∞)- star ), then h ( P ) can be arbitrarily large. For each P * in P 5 with neither vertices of the degree from 6 to 8 nor minor (5, 5, 5, 5,∞)-star, it follows from Lebesgue’s Theorem that h ( P * ) ≤ 17. We prove in particular that every such polytope P * satisfies h ( P * ) ≤ 12, 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 P 5 under the absence of minor (5, 5, 5, 5,∞)- stars can reach 15, 17, or 14, respectively.