Extended 5-Cycle Systems

An extended 5-cycle system of order \(n\) is an ordered pair \((V, B)\), where \(B\) is a collection of edge-disjoint 5-cycles, 2-tadpoles, and loops that partition the edges of the graph \(K_n^+\) whose vertex set is an \(n\)-set \(V\). In this paper, we show that an extended 5-cycle system of order \(n\) exists for all \(n\) except \(n = 2\) and \(3\).