Disjoint cycles covering specified vertices in bipartite graphs with partial degrees

Abstract

Let G be a balanced bipartite graph of order 2n with bipartition (X,Y), and S a subset of X. Suppose that every pair of nonadjacent vertices {x,y} with x∈S,y∈Y satisfies dG(x)+dG(y)≥n+1. We show that if |S|≥2k+1, then G contains k disjoint cycles such that each of them contains at least two vertices of S. Moreover, if |S|≥2k+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.