Extended $7$-cycle Systems

Abstract

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 $5$.