The following set of drawings pertain to the Hoffman-Singleton graph,
a.k.a., the Moore graph of degree 7 and diameter 2, the (7,5)-cage,
and the strongly regular graph with parameters (50,7,0,1).
Any vertex v in this graph has 7 neighbors. Each of the 7 neighbors
is adjacent to 6 other vertices (accounting for all 50 vertices).
Call the 7 sets of size 6 ** the I-sets for v**.
The I-sets are disjoint, lest there be
4-cycles in the graph; they are independent, lest there be triangles.
Any two I-sets (for the same vertex) induce a matching. Any three I-sets induce 3 copies of
a 6-cycle. The figures show the graphs induced by larger numbers of
I-sets.