On Quadrilaterals in a Bipartite Graph

Abstract

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 $d(x)+d(y)\ge 3k$ for every pair of nonadjacent vertices $x\in V_1$, $y\in V_2$, then $G$ contains $k$ independent quadrilaterals.