Ore-type conditions for bipartite graphs containing hexagons

Let G = ( V 1 , V 2 ; E ) be a bipartite graph with | V 1 | = | V 2 | = 3 k , where k > 0 . In this paper it is proved that if d ( x ) + d ( y ) ≥ 4 k − 1 for every pair of nonadjacent vertices x ∈ V 1 , y ∈ V 2 , then G contains k − 1 independent cycles of order 6 and a path of order 6 such that all of them are independent. Furthermore, if d ( x ) + d ( y ) ≥ 4 k for every pair of nonadjacent vertices x ∈ V 1 , y ∈ V 2 and k > 2 , then G contains k − 2 independent cycles of order 6 and a cycle of order 12 such that all of them are independent.