Weakly pancyclic graphs

  • Authors:

  • Stephan Brandt;Ralph Faudree;Wayne Goddard

  • Affiliations:

  • FB Mathematik, Freie Universität Berlin, Graduiertenkolleg ‘Algorithmische Diskrete Mathematik’, Arnimallee 2-6, 14195 Berlin, Germany;Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152;Department of Computer Science, University of Natal, Durban 4041, South Africa

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 1998

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Abstract

In generalizing the concept of a pancyclic graph, we say that a graph is “weakly pancyclic” if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic. This sheds some light on the content of a famous metaconjecture of Bondy. From the main result of this paper it follows that 2-connected nonbipartite graphs of sufficiently large order n with minimum degree exceeding 2n-7 are weakly pancyclic; and that graphs with minimum degree at least n-4 + 250 are pancyclic, if they contain both a triangle and a hamiltonian cycle. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 141–176, 1998