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In 1940, in attempts to solve the Four Color Problem, H. Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P 5 of 3-polytopes with minimum degree 5. Given a 3-polytope P , by h ( P ) we denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P . In 1996, S. Jendrol’ and T. Madaras showed that if a...
Let w Δ be the minimum integer W with the property that every 3-polytope with minimum degree 5 and maximum degree Δ has a vertex of degree 5 with the degree-sum (weight) of all vertices in its closed neighborhood at most W . Trivially, w 5 = 30 and w 6 = 35 . In 1940, Lebesgue proved w Δ ≤ Δ + 31 for all Δ ≥ 5 and w Δ ≤ Δ + 27 for Δ ≥ 41 . In 1998, the...
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