Essential independent sets and long cycles

  • Authors:

  • Kazuhide Hirohata

  • Affiliations:

  • Department of Mathematics, Kieo University 3-14-1, Hiyoshi,kohoku-ku Yokohama, 223-8522, Japan

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

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Abstract

An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. For S ⊆ V(G) with S ≠ φ, let Δ(S) = max{dG(x)|x ∈ S}. We prove the following theorem. Let k ≥ 2 and let G be a k-connected graph. Suppose that Δ(S) ≥ d for every essential independent set S of order k. Then G has a cycle of length at least min{|G|,2d}. This generalizes a result of Fan.