Light Minor 5-Stars in 3-Polytopes with Minimum Degree 5

Resumo

Attempting to solve the Four Color Problem in 1940, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P₅ of 3-polytopes with minimum degree 5. This description depends on 32 main parameters. Not many precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in P₅. Given a 3-polytope P, by w(P) denote the minimum of the maximum degree-sum (weight) of the neighborhoods of 5-vertices (minor 5-stars) in P. In 1996, Jendrol’ and Madaras showed that if a polytope P in P₅ is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5, ∞)-star), then w(P) can be arbitrarily large. For each P* in P₅ with neither vertices of degree 6 and 7 nor minor (5, 5, 5, 5, ∞)-star, it follows from Lebesgue’s Theorem that w(P*) ≤ 51. We prove that every such polytope P* satisfies w(P*) ≤ 42, which bound is sharp. This result is also best possible in the sense that if 6-vertices are allowed but 7-vertices forbidden, or vice versa; then the weight of all minor 5-stars in P₅ under the absence of minor (5, 5, 5, 5, ∞)-stars can reach 43 or 44, respectively.