Forbidden Subgraphs for Hamiltonicity of 3-Connected Claw-Free Graphs


  • Contract grant sponsor: JSPS Grant-in-Aid for Young Scientists (B); Contract grant number: 22740068, 2010


In this article, we consider forbidden subgraphs for hamiltonicity of 3-connected claw-free graphs. Let math formula be the graph obtained from a triangle by attaching a path of length i to one of its vertices, and let math formula be the graph obtained from the Petersen graph by adding one pendant edge to each vertex. Lai et al. (J Graph Theory 64(1) (2010), 1–11) conjectured that every 3-connected math formula-free graph G is hamiltonian unless G is the line graph of math formula. It is shown in this article that this conjecture is true. Moreover, we investigate the set of connected graphs math formula which satisfies that every 3-connected math formula-free graph of sufficiently large order is hamiltonian if and only if A is a member of math formula. We prove that, if math formula, then G is a graph on at most 12 vertices with the following structure: (i) a path of length at most 10, (ii) a triangle with three vertex-disjoint paths of total length at most 9, or (iii) G consists of two triangles connected by a path of length 1, 3, 5, or 7. AMS classification: 05C45, 05C38, 05C75. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 146–160, 2013