Disjoint cycles covering specified vertices in bipartite graphs with partial degrees

Let G be a balanced bipartite graph of order 2 n with bipartition (X, Y ) , and S a subset of X. Suppose that every pair of nonadjacent vertices {x,y} with x E S,y E Y satisfies dG(x) + dG (y ) >= n + 1 . We show that if | S| >= 2 k + 1 , then G contains k disjoint cycles such that each of them contains at least two vertices of S. Moreover, if | S| >= 2 k + 2 , then G contains k disjoint cycles covering S such that each of them contains at least two vertices of S. Here, the lower bounds of | S| are necessary, and for the latter result the degree condition is sharp.(c) 2022 Elsevier Inc. All rights reserved.