Claw-free graphs. I. Orientable prismatic graphs

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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 languageEnglish (US)
Pages (from-to)867-903
Number of pages37
JournalJournal of Combinatorial Theory. Series B
Issue number6
StatePublished - Nov 2007

All Science Journal Classification (ASJC) codes

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


  • Claw-free graph
  • Prismatic
  • Structure theorem


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