Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2002-05

May 2002

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

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


This paper is a continuation of Part I: Triangulations, LSE-CDAM-2002-04.

Given a plane graph, a k-star at u is a set of k vertices w ith 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 $d\leq5$ 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 $\Delta\geq47$ the chromatic number of the square of G is at most $\lceil\frac95\,\Delta\rceil+1$. This improves existing bounds on the chromatic number of the square of a planar graph.

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