**Originator(s):**

**Definitions:**
The *Hoffman-Singleton graph* [HS] is a 7-regular graph with girth 5 on
50 vertices. It is the only such graph with the fewest vertices, which makes
it the unique (7,5)-cage. (Other proofs of uniqueness appear in [FS, OW].)
The graph consists of ten 5-cycles *P _{0},...,P_{4}* and

**Question:**
Does *K _{50}* decompose into 7 copies of the Hoffman-Singleton
graph?

**Comments/Partial results:**
Meszka and Siagiova [MS] have found five edge-disjoint copies of the
Hoffman-Singleton graph in *K _{50}*, using voltage graph methods.

**References:**

[FS] Fan, C.; Schwenk, A. J.
Structure of the Hoffman-Singleton graph. Proceedings of the Twenty-fourth
Southeastern International Conference on Combinatorics, Graph Theory, and
Computing (Boca Raton, FL, 1993). Congr. Numer. 94 (1993), 3--8.

[HS] Hoffman, A. J.; Singleton, R. R.
On Moore graphs with diameters 2 and 3.
IBM J. Res. Develop. 4 1960 497--504.

[MS] Meszka, M.; Siagiova, J.
A covering construction for packing disjoint copies
of the Hoffman-Singleton graph into *K _{50}*.
J. Combinatorial Designs, to appear.

[OW] O'Keefe, M.; Wong, P. K. On certain regular graphs of girth 5. Internat. J. Math. Math. Sci. 7 (1984), no. 4, 785--791.