Essential independent sets and long cycles

doi:10.1016/S0012-365X(01)00277-1

Discrete Mathematics

Volume 250, Issues 1–3, 6 May 2002, Pages 109-123

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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≠∅, let Δ(S)=max{d_G(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.

Keywords

Cycle

Length

Essential independent set

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¹: Present address: Department of Electronic and Computer Engineering, Ibaraki National College of Technology, 866 Nakane Hitachinaka-shi, Ibaraki-ken 312-8508 Japan.