Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200204May 2002 
O. V. Borodin, H. J. Broersma, A. Glebov, and J. van den Heuvel
Abstract
Given a plane graph, a kstar 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 (d1)star centered at a vertex of degree d at most 5 and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch.
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