On 2-factors with cycles containing specified edges in a bipartite graph

Let k ≥ 1 be an integer and G = ( V 1 , V 2 ; E ) a bipartite graph with | V 1 | = | V 2 | = n such that n ≥ 2 k + 2 . In this paper it has been proved that if for each pair of nonadjacent vertices x ∈ V 1 and y ∈ V 2 , d ( x ) + d ( y ) ≥ ⌈ 4 n + 2 k − 1 3 ⌉ , then for any k independent edges e 1 , … , e k of G , G has a 2-factor with k + 1 cycles C 1 , … , C k + 1 such that e i ∈ E ( C i ) and | V ( C i ) | = 4 for each i ∈ { 1 , … , k } . We shall also show that the conditions in this paper are sharp.