On Quadrilaterals in a Bipartite Graph

Let \( G = (V_1, V_2; E) \) be a bipartite graph with \( |V_1| = |V_2| = 2k \), where \( k \) is a positive integer. It is proved that if \( d(x) + d(y) \geq 3k \) for every pair of nonadjacent vertices \( x \in V_1 \), \( y \in V_2 \), then \( G \) contains \( k \) independent quadrilaterals.