Fan-Type Theorem for Long Cycles Containing a Specified Edge

In this paper, we prove that if G is 3-connected noncomplete graph of order n satisfying min{max{d(u), d(v)} : d(u, v) = 2} = µ, then for each edge e, G has a cycle containing e of length at least min{n, 2µ}, unless G is a spanning subgraph of Kµ + K c n−µ or K3 + (lKµ−2 ∪ Ks), where n = l(µ − 2) + s + 3, 1 ≤ s ≤ µ − 2