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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 1n9, with two genuine exceptions.
05C70Factorization, etc.