Claw-free graphs. I. Orientable prismatic graphs

Authors:
Maria Chudnovsky;Paul Seymour
Affiliations:
Columbia University, New York, NY 10027, USA;Princeton University, Princeton, NJ 08544, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

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Abstract

A graph is prismatic if for every triangle T, every vertex not in T has exactly one neighbour in T. In this paper and the next in this series, we prove a structure theorem describing all prismatic graphs. This breaks into two cases depending whether the graph is 3-colourable or not, and in this paper we handle the 3-colourable case. (Indeed we handle a slight generalization of being 3-colourable, called being ''orientable.'') Since complements of prismatic graphs are claw-free, this is a step towards the main goal of this series of papers, providing a structural description of all claw-free graphs (a graph is claw-free if no vertex has three pairwise nonadjacent neighbours).