Bi-cycle extendable through a given set in balanced bipartite graphs

Let G=(X,Y;E)G=(X,Y;E) be a balanced bipartite graph of order 2n2n. The path-cover number  pc(H)pc(H) of a graph HH is the minimum number of vertex-disjoint paths that use up all the vertices of HH. S⊆V(G)S⊆V(G) is called a balanced set of GG if |S∩X|=|S∩Y||S∩X|=|S∩Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph GG satisfying that for every balanced set SS, there is a bi-cycle of every length from |S|+2pc(〈S〉)|S|+2pc(〈S〉) up to 2n2n through SS.