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

In two recent papers [Hafner, Elec. J. Combin. 11 (R77) (2004), 1–33; Ilić, Pace and Magliveras, J. Combin. Math. Combin. Comput. 80 (2012), 267–275] it was shown that the Higman-Sims graph Γ 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 Γ. In this article we prove that the Hall-Janko graph Θ 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 J2 = HJ . The stabilizer in J2 of such a decomposition is D5 × A5. There are striking similarities between the decompositions of Γ and Θ 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.