Super-Simple Twofold Steiner Pentagon Systems.

A twofold pentagon system of order v is a decomposition of the complete undirected 2-multigraph 2 K into pentagons. A twofold Steiner pentagon system of order v [TSPS( v)] is a twofold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. A TSPS( v) is said to be super-simple if its underlying ( v, 5, 4)-BIBD is super-simple; that is, if any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary conditions for the existence of a super-simple TSPS( v); namely, v ≥ 15 and v ≡ 0 or 1 ( mod 5) are sufficient. For these specified orders, the main result of this paper also guarantees the existence of a very special and interesting class of twofold and fourfold Steiner pentagon systems of order v with the additional property that, for any two vertices, the two or four paths of length two joining them are distinct. [ABSTRACT FROM AUTHOR]