Related decompositions and new constructions of the Higman-Sims and Hall-Janko graphs

In two recent papers [Hafner, Elec. J. Combin. 11 (R77) (004), 1-33; Ilic, Pace and Magliveras, J. Combin. Math. Combin. Comput. 80 (2012), 267-275] it was shown that the Higman-Sims graph Gamma can be decomposed into a disjoint union of five double Petersen graphs. In the second of these papers, it was further shown that all such decompositions fall into a single orbit under the action of sporadic simple group HS, which is of index two in the full automorphism group of G. In this article we prove that the Hall-Janko graph Theta can be decomposed into a disjoint union of double co-Petersen graphs. We find all such decompositions, and prove they fall into a single orbit under the action of the sporadic simple group J(2) - HJ. The stabilizer in J(2) of such a decomposition is D-5 x A(5). There are striking similarities between the decompositions of G and Theta just described. Finally, motivated by these decompositions, we obtain new constructions of the Higman-Sims and Hall-Janko graphs from Petersen and co-Petersen graphs.