Elsevier

Discrete Mathematics

Volume 339, Issue 2, 6 February 2016, Pages 923-930
Discrete Mathematics

Low stars in normal plane maps with minimum degree 4 and no adjacent 4-vertices

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Abstract

We consider normal plane maps M4 with minimum degree at least 4 and no adjacent 4-vertices. The height of a star is the maximum degree of its vertices. By h(Sk) and h(Sk(m)) with 1k4 we denote the minimum height of arbitrary k-stars and k-stars centered at vertices of degree at most 5, respectively, in a given M4.

Mohar, Škrekovski, and Voss proved (2003) that every M4 satisfies h(S4)107. We improve this result by proving that h(S4)23 and construct an M4 with h(S4)=18. On the other hand, we show that h(S4(m))=.

Also, we prove that every M4 satisfies h(S3)10 and h(S3(m))11, where both 10 and 11 are sharp.

Keywords

Plane map
Plane graph
3-polytope
Structural property
Star
Weight
Height

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