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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 ={𝒫(x):xV(X)} of subgraphs of X, all isomorphic with G, such that

every edge of X occurs in exactly two members of and

𝒫(x) and 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:
05C25Graphs and abstract algebra
References:
[1]El-Shanawany, R.; Gronau, H. -D.O.F.; Grüttmüller, M.: Orthogonal double covers of kn,n by small graphs, Discrete applied mathematics 138, 47-63 (2004) · Zbl 1034.05039 · doi:10.1016/S0166-218X(03)00269-5
[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)
[3]Hartmann, S.; Schumacher, U.: Orthogonal double covers of general graphs, Discrete applied mathematics 138, 107-116 (2004) · Zbl 1034.05040 · doi:10.1016/S0166-218X(03)00274-9
[4]Lauri, J.; Scapellato, R.: Topics in graph automorphisms and reconstruction, London math. Soc. S.T., vol. 54, (2003)