On Quadrilaterals and Cycle Covers in a Bipartite Graph

In [13], we conjectured that if \(G = (V_1, V_2; E)\) is a bipartite graph with \(|V_1| = |V_2| = 2k\) and minimum degree at least \(k + 1\), then \(G\) contains \(k\) vertex-disjoint quadrilaterals. In this paper, we propose a more general conjecture: If \(G = (V_1, V_2; E)\) is a bipartite graph such that \(|V_1| = |V_2| = n \geq 2\) and \(\delta(G) \geq [n/2] + 1\), then for any bipartite graph \(H = (U_1, U_2; F)\) with \(|U_1| \leq n, |U_2| \leq n\) and \(\Delta(H) \leq 2, G\) contains a subgraph isomorphic to \(H\). To support this conjecture, we prove that if \(n = 2k + t\) with \(k \geq 0\) and \(t \geq 3, G\) contains \(k + 1\) vertex-disjoint cycles covering all the vertices of \(G\) such that \(k\) of them are quadrilaterals.