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All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5

100%

Borodin O. V., Ivanova A. O.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 1

5-12

Lebesgue (1940) proved that every 3-polytope P5 of girth 5 has a path of three vertices of degree 3. Madaras (2004) refined this by showing that every P5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P5 in which every 3-vertex has a 4-neighbor. We give another tight description of 3-stars in P5s: there is a vertex of degree at most 4 having three 3-neighbors. Furthermore, we show that there are only these two tight descriptions of 3-stars in P5s. Also, we give a tight description of stars with at least three rays in P5s and pose a problem of describing all such descriptions. Finally, we prove a structural theorem about P5s that might be useful in further research.

On the Weight of Minor Faces in Triangle-Free 3-Polytopes

88%

Borodin O. V., Ivanova A. O.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 3

603-619

The weight w(f) of a face f in a 3-polytope is the degree-sum of vertices incident with f. It follows from Lebesgue’s results of 1940 that every triangle-free 3-polytope without 4-faces incident with at least three 3-vertices has a 4-face with w ≤ 21 or a 5-face with w ≤ 17. Here, the bound 17 is sharp, but it was still unknown whether 21 is sharp. The purpose of this paper is to improve this 21 to 20, which is best possible.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Borodin O. V.

2 Ivanova A. O.

1 2017

1 2016

Wyniki wyszukiwania

All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5

On the Weight of Minor Faces in Triangle-Free 3-Polytopes