Extended 7-cycle Systems

An extended 7-cycle system of order \( n \) is an ordered pair \( (V, B) \), where \( B \) is a collection of edge-disjoint 7-cycles, 3-tadpoles, and loops which partition the edges of the graph \( K_n^+ \) whose vertex set is an \( n \)-set \( V \). In this paper, we show that an extended 7-cycle system of order \( n \) exists for all \( n \) except \( n = 2, 3, \) and \( 5 \).