Ore-type conditions for bipartite graphs containing hexagons

Let G=(V1,V2;E)G=(V1,V2;E) be a bipartite graph with |V1|=|V2|=3k|V1|=|V2|=3k, where k>0k>0. In this paper it is proved that if d(x)+d(y)≥4k−1d(x)+d(y)≥4k−1 for every pair of nonadjacent vertices x∈V1x∈V1, y∈V2y∈V2, then GG contains k−1k−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)≥4kd(x)+d(y)≥4k for every pair of nonadjacent vertices x∈V1x∈V1, y∈V2y∈V2 and k>2k>2, then GG contains k−2k−2 independent cycles of order 6 and a cycle of order 12 such that all of them are independent.