Vertex-Disjoint Quadrilaterals in Multigraphs

A cycle of length four is called a quadrilateral and a multigraph is called standard if every edge in it has multiplicity at most two. We prove that if M is a standard multigraph of order 4k, where k is a positive integer and the minimum degree of M is at least , then M contains k vertex-disjoint quadrilaterals, such that each quadrilateral contains at least three multiedges, with only two exceptions. This implies the main result obtained by Zhang and Wang [J Graph Theory 50:91-104, 2005]: Let D be a directed graph of order 4k, where k is a positive integer. Suppose that the minimum degree of D is at least , then D contains k vertex-disjoint directed quadrilaterals with only one exception.