# Stars and bunches in planar graphs. Part I: Triangulations

O.V. Borodin, H.J. Broersma, A. Glebov, J. van den Heuvel

Research output: Book/ReportReportProfessional

## Abstract

Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We prove a theorem on the structure of plane triangulations in terms of stars and bunches. The result states that a plane triangulation contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch.
Original language English Enschede University of Twente, Department of Applied Mathematics 18 Published - 2002

### Publication series

Name Memorandum University of Twente, Department of Applied Mathematics 1632 0169-2690

• MSC-05C12
• MSC-05C15

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