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Keywords:

  • edge colorings;
  • critical graphs;
  • 2-factors;
  • Hamiltonian cycles

Abstract

In this article, we consider Vizing's 2-Factor Conjecture which claims that any Δ-critical graph has a 2-factor, and show that if G is a Δ-critical graph with n vertices satisfying inline image, then G is Hamiltonian and thus G has a 2-factor. Meanwhile in this article, we also consider long cycles of overfull critical graphs and obtain that if G is an overfull Δ-critical graph with n vertices, then the circumference of G is at least mininline image.© 2012 Wiley Periodicals, Inc. J. Graph Theory 00: 1-14, 2012