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

Abstract

Let $k$, $n$ be integers with $k\ge 3$ and $n\ge 3k$, and let $G$ be a bipartite graph having partite sets $V_1$, $V_2$ with $\left|V_1\right|=\left|V_2\right|=n$. We show that if $d_G(u)+d_G(\upsilon )\ge n+2k-1$ for any $u\in V_1$ and $\upsilon \in V_2$ with $u\upsilon \cancel{\in }E(G)$, then for any vertex disjoint paths $P_1,P_2,\dots ,P_k$ of order 3, $G$ contains vertex disjoint cycles $H_1,H_2,\mathrm{...},H_k$ such that ${{\displaystyle \cup }}_{1\le i\le k}V(H_i)=V(G)$ and $H_i$ passes through $P_i$ for each $i$ with $1\le i\le k$.