Describing the neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 6 to 8

Abstract In 1940, Lebesgue proved that every 3-polytope with minimum degree 5 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: ( 6 , 6 , 7 , 7 , 7 ) , ( 6 , 6 , 6 , 7 , 9 ) , ( 6 , 6 , 6 , 6 , 11 ) , ( 5 , 6 , 7 , 7 , 8 ) , ( 5 , 6 , 6 , 7 , 12 ) , ( 5 , 6 , 6 , 8 , 10 ) , ( 5 , 6 , 6 , 6 , 17 ) , ( 5 , 5 , 7 , 7 , 13 ) , ( 5 , 5 , 7 , 8 , 10 ) , ( 5 , 5 , 6 , 7 , 27 ) , ( 5 , 5 , 6 , 6 , ∞ ) , ( 5 , 5 , 6 , 8 , 15 ) , ( 5 , 5 , 6 , 9 , 11 ) , ( 5 , 5 , 5 , 7 , 41 ) , ( 5 , 5 , 5 , 8 , 23 ) , ( 5 , 5 , 5 , 9 , 17 ) , ( 5 , 5 , 5 , 10 , 14 ) , ( 5 , 5 , 5 , 11 , 13 ) . Recently, we proved that forbidding vertices of degree from 7 to 11 results in a tight description ( 5 , 5 , 6 , 6 , ∞ ) , ( 5 , 6 , 6 , 6 , 15 ) , ( 6 , 6 , 6 , 6 , 6 ) . The purpose of this note is to prove that every 3-polytope with minimum degree 5 and no vertices of degrees 6, 7, and 8 has a 5-vertex whose neighborhood is majorized by one of the sequences ( 5 , 5 , 5 , 5 , ∞ ) and ( 5 , 5 , 5 , 10 , 12 ) , which is tight and improves a corresponding description ( 5 , 5 , 5 , 5 , ∞ ) , ( 5 , 5 , 9 , 5 , 17 ) , ( 5 , 5 , 10 , 5 , 14 ) , ( 5 , 5 , 11 , 5 , 13 ) that follows from the Lebesgue Theorem.