Low 5-stars at 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 7 to 9

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 k...

5-stars of Low Weight in Normal Plane Maps with Minimum Degree 5

It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48.

The number of vertices of degree 5 in a contraction-critically 5-connected graph

An edge of a k-connected graph is said to be k-contractible if its contraction yields a k-connected graph. A non-complete k-connected graph possessing no k-contractible edges is called contraction-critical k-connected. Let G be a contraction-critical 7-connected graph with n vertices, and V7 the set of vertices of degree 7 in G. In this paper, we prove that |V7| ≥ n 22 , which improves the resu...

Nonseparating vertices in tournaments with large minimum degree

N ov 2 00 5 Counting d - polytopes with d + 3 vertices

We completely solve the problem of enumerating combinatorially inequivalent d-dimensional polytopes with d + 3 vertices. A first solution of this problem, by Lloyd, was published in 1970. But the obtained counting formula was not correct, as pointed out in the new edition of Grünbaum’s book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution, relying on...