Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs

  • Authors:

  • Zdeněk Ryjáček;Petr Vrána

  • Affiliations:

  • Department of Mathematics, University of West Bohemia and Institute for Theoretical Computer Science (ITI) Charles University, P.O. Box 314 306 14 Pilsen, Czech Republic;Department of Mathematics, University of West Bohemia and Institute for Theoretical Computer Science (ITI) Charles University, P.O. Box 314 306 14 Pilsen, Czech Republic

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2011

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Abstract

We introduce a closure concept that turns a claw-free graph into the line graph of a multigraph while preserving its (non-)Hamilton-connectedness. As an application, we show that every 7-connected claw-free graph is Hamilton-connected, and we show that the well-known conjecture by Matthews and Sumner (every 4-connected claw-free graph is hamiltonian) is equivalent with the statement that every 4-connected claw-free graph is Hamilton-connected. Finally, we show a natural way to avoid the non-uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:152-173, 2011 © 2011 Wiley Periodicals, Inc.