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Orthogonal double covers of Cayley graphs. (English) Zbl 1213.05128

Summary: Let \(X\) and \(G\) be graphs, such that \(G\) is isomorphic to a subgraph of \(X\).
An orthogonal double cover (ODC) of \(X\) by \(G\) is a collection \(\mathcal B = \{\mathcal P(x) : x \in V(X)\}\) of subgraphs of \(X\), all isomorphic with \(G\), such that
(i)
every edge of \(X\) occurs in exactly two members of \(\mathcal B\) and
(ii)
\(\mathcal P(x)\) and \(\mathcal P(y)\) share an edge if and only if \(x\) and \(y\) are adjacent in \(X\).
The main question is: given the pair \((X,G)\), is there an ODC of \(X\) by \(G\)? An obvious necessary condition is that \(X\) is regular.
A technique to construct ODCs for Cayley graphs is introduced. It is shown that for all \((X,G)\) where \(X\) is a 3-regular Cayley graph on an abelian group there is an ODC, a few well known exceptions apart.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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References:

[1] El-Shanawany, R.; Gronau, H.-D. O.F.; Grüttmüller, M., Orthogonal double covers of \(K_{n, n}\) by small graphs, Discrete Applied Mathematics, 138, 47-63 (2004) · Zbl 1034.05039
[2] Gronau, H.-D. O.F.; Hartmann, S.; Grüttmüller, M.; Leck, U.; Leck, V., On orthogonal double covers of graphs, Design Codes Cryptography, 27, 49-91 (2002) · Zbl 1001.05091
[3] Hartmann, S.; Schumacher, U., Orthogonal double covers of general graphs, Discrete Applied Mathematics, 138, 107-116 (2004) · Zbl 1034.05040
[4] Lauri, J.; Scapellato, R., Topics in Graph Automorphisms and Reconstruction, London Math. Soc. S.T., Vol. 54 (2003), Cambridge University Press
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