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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 $\cal B = \{\cal P(x) : x \in V(X)\}$ of subgraphs of $X$, all isomorphic with $G$, such that {\parindent=6mm \item{(i)}every edge of $X$ occurs in exactly two members of $\cal B$ and \item{(ii)}$\cal P(x)$ and $\cal P(y)$ share an edge if and only if $x$ and $y$ are adjacent in $X$. \par} 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.

05C25Graphs and abstract algebra
Full Text: DOI
[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) · Zbl 1038.05025