Embedding 5-cycle systems into pentagon triple systems (Q1044955)

A \textit{pentagon triple} is a graph created from a 5-cycle by replacing each edge by a triangle. Vertices that are common to neighboring triangles induce a 5-cycle called \textit{inside} 5-cycle. A \textit{pentagon triple system} is a partition of a complete graph \(K_n\) into pentagon triples. Authors show that the spectrum for pentagon triple systems is the set of all \(n\equiv\) 1, 15, 21 or \(25\pmod{30}\). A \textit{5-cycle system} of order \(n\) is a partition of complete graph \(K_n\) into 5-cycles. Authors construct a 5-cycle systems of order \(10m+1\) and \(10m+5\) that can be embedded into pentagon triple systems of order \(30m+1\) and \(30m+15\) as inside cycles, respectively. The possible exception is a 5-cycle system of order 21 and pentagon triple system of order 61.