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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 {\it J. H. Dinitz} and {\it D. R. Stinson} (eds.) [Contemporary design theory (Wiley, New York) (1992; Zbl 0746.00028), pp. 13--40 (Chapter 2)]; {\it H.-J. O. F. Gronau} et al. [Des. Codes Cryptography 27, 49--91 (2002; Zbl 1001.05091)]; {\it H.-J. O. F. Gronau} et al. [Graphs Comb. 13, 251--262 (1997; Zbl 0885.05093)]; {\it 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 $1 \leqslant n \leqslant 9$, with two genuine exceptions.

##### MSC:
 05C70 Factorization, etc.
##### Keywords:
Orthogonal double cover; ODC; Graph decompositions
nauty
Full Text:
##### References:
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