Stars and bunches in planar graphs. Part I: Triangulations

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 ≤ 5 and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch. ∗This is a translated and adapted version of a paper that appeared in Diskretn. Anal. Issled. Oper. Ser. 1 8 (2001) no. 2, 15–39 ( in Russian ). The first three authors are supported by NWO grant 047-008-006.