Steiner pentagon covering designs (Q5937573)

A pentagon system (PS) of order \(n\) is a decomposition of the complete graph \(K_n\) into edge disjoint pentagons. A Steiner pentagon system (SPS) is a PS with the additional property that every pair of vertices is joined by a path of length 2 in exactly one pentagon. If we relax the conditions and ask for a set of pentagons such that every edge occurs in at least one pentagon and every pair of vertices is joined by a path of length 2 in at least one pentagon, then we get a Steiner pentagon covering (SPC). The SPCs that contain (the minimum number of) \(\lceil n/5 \lceil (n-1)/2 \rceil \rceil\) pentagons are called Steiner pentagon covering designs (SPCDs). In the current paper, the existence of SPCDs is studied. For \(n\) even, the existence of SPCDs is established with a few possible exceptions. For \(n\) odd, several new SPCDs are found. Some results on the related packing problem are also obtained.