Vertex disjoint cycles containing specified paths of order 3 in a bipartite graph

Let k, n be integers with k≥ 3 and n≥3k, and let G be a bipartite graph having partite sets V1, V2 with |V1|=|V2|=n. We show that if dG(u)+dG(υ)≥n+2k−1 for any u∈V1 and υ∈V2 with uυ∈E(G), then for any vertex disjoint paths P1,P2,…,Pk of order 3, G contains vertex disjoint cycles H1,H2,...,Hk such that ∪1≤i≤k V(Hi)=V(G) and Hi passes through Pi for each i with 1≤i≤k.