Stars and bunches in planar graphs. Part II: General planar graphs and colourings

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 first prove a theorem on the structure of plane graphs in terms of stars and bunches. The result states that a plane graph 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 structural result is used to prove a best possible upper bound on the minimum degree of the square of a planar graph, and hence on a best possible bound for the number of colours needed in a greedy colouring of it. In particular, we prove that for a planar graph G with maximum degree ∆ ≥ 47 the chromatic number of the square of G is at most d 9 5 ∆e + 1. This improves existing bounds on the chromatic number of the square of a planar graph. ∗This is a translated and adapted version of a paper that appeared in Diskretn. Anal. Issled. Oper. Ser. 1 8 (2001) no. 4, 9–33 ( in Russian ). The first three authors are supported by NWO grant 047-008-006.