An orthogonal double cover (ODC) of Kn is a collection of graphs such that each edge of Kn occurs in exactly two of the graphs and two graphs have precisely one edge in common. ODCs of Kn and their generalizations have been extensively studied by several authors (e.g. in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory, Wiley, New York, 1992, pp. 13–40 (Chapter 2); Design Codes Cryptography 27 (2002) 49; Graphs Combin. 13 (1997) 251; V. Leck, Orthogonal double covers of Km, Ph.D. Thesis, Universität Rostock, 2000). In this paper, we investigate ODCs where the graph to be covered twice is Kn,n and all graphs in the collection are isomorphic to a given small graph G. We prove that there exists an ODC of Kn,n by all proper subgraphs G of Kn,n for 1⩽n⩽9, with two genuine exceptions.
