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On the structural result on normal plane maps

100%

Madaras T., Marcinová A.

Discussiones Mathematicae Graph Theory

2002

tom 22

nr 2

293-303

We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by $6 + [(2D+12)/(D-2)]((D-1)^{(t-1)} - 1)$. This improves a recent bound $6 + [(3D+3)/(D-2)]((D-1)^{t-1}-1)$, D ≥ 8 by Jendrol' and Skupień, and the upper bound for distance-2 chromatic number.

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

70%

Borodin O. V., Ivanova A. O., Jensen T. R.

Discussiones Mathematicae Graph Theory

2014

tom 34

nr 3

539-546

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

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

70%

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

3 Discussiones Mathematicae Graph Theory

2 Borodin O. V.

2 Ivanova A. O.

1 Jensen T. R.

1 Madaras T.

1 Marcinová A.

1 2016

1 2014

1 2002

Wyniki wyszukiwania

On the structural result on normal plane maps

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

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