## 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).

Original language | English (US) |
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Pages (from-to) | 867-903 |

Number of pages | 37 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 97 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2007 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Keywords

- Claw-free graph
- Prismatic
- Structure theorem