Abstract
In this article, we consider forbidden subgraphs for hamiltonicity of 3-connected claw-free graphs. Let be the graph obtained from a triangle by attaching a path of length i to one of its vertices, and let 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 -free graph G is hamiltonian unless G is the line graph of . It is shown in this article that this conjecture is true. Moreover, we investigate the set of connected graphs which satisfies that every 3-connected -free graph of sufficiently large order is hamiltonian if and only if A is a member of . We prove that, if , 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
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.