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Orthogonal double covers of ${K}_{n,n}$ by small graphs. (English) Zbl 1034.05039
Summary: An orthogonal double cover (ODC) of ${K}_{n}$ is a collection of graphs such that each edge of ${K}_{n}$ occurs in exactly two of the graphs and two graphs have precisely one edge in common. ODCs of ${K}_{n}$ and their generalizations have been extensively studied by several authors (e.g. in J. H. Dinitz and D. R. Stinson (eds.) [Contemporary design theory (Wiley, New York) (1992; Zbl 0746.00028), pp. 13–40 (Chapter 2)]; H.-J. O. F. Gronau et al. [Des. Codes Cryptography 27, 49–91 (2002; Zbl 1001.05091)]; H.-J. O. F. Gronau et al. [Graphs Comb. 13, 251–262 (1997; Zbl 0885.05093)]; V. Leck [Orthogonal double covers of ${K}_{m}$, Ph.D. Thesis, Universität Rostock (2000)]). In this paper, we investigate ODCs where the graph to be covered twice is ${K}_{n,n}$ and all graphs in the collection are isomorphic to a given small graph $G$. We prove that there exists an ODC of ${K}_{n,n}$ by all proper subgraphs $G$ of ${K}_{n,n}$ for $1⩽n⩽9$, with two genuine exceptions.
##### MSC:
 05C70 Factorization, etc.
##### Keywords:
Orthogonal double cover; ODC; Graph decompositions