An extension of a theorem on cycles containing specified independent edges (Q1348118)

The authors give an alternative proof of the following theorem of Egawa et al.: If \(k\) is an integer \(\geq 2\) and \(G\) is a graph of order \(n \geq 4k-1\) in which \(d(u)+d(v) \geq n+2k-2\) for every pair of non-adjacent vertices \(u\) and \(v\), then for any \(k\) independent edges \(e_1, \ldots ,e_k\) of \(G\), there exist \(k\) vertex-disjoint cycles \(C_1, \ldots ,C_k\) in \(G\) such that (i) \(e_i \in E(C_i)\) for all \(1 \leq i \leq k\), and (ii) \(V(C_1) \cup \cdots \cup V(C_k)=V(G)\). In their proof, they produce cycles which with the possible exception of one are all of length 3 or 4.