An extension of a theorem on cycles containing specified independent edges Original Research Article

We give an alternative proof of a conjecture due to Wang (J. Graph Theory 26 (1997) 105) in a stronger form. The main theorem states that for any integer k⩾2 if G is a graph of order n⩾4k−1 and d(u)+d(v)⩾n+2k−2 for each pair of non-adjacent vertices u and v of G, then, for any k independent edges e1,…,ek of G, there exist k vertex-disjoint cycles C1,…,Ck in G such that (i) ei∈E(Ci) for all 1⩽i⩽k, (ii) V(C1)∪⋯∪V(Ck)=V(G), and (iii) #{i⩽k | |Ci|>4}⩽1, unless Ka+K̄2k+Kn−2k−a⊂G⊂Ka+K2k+Kn−2k−a for some a (2k−2