Path Decompositions Which Contain No Proper Subsystems

Abstract

A $G$-design is a partition of $E(K_v)$ in which each element induces a copy of $G$. The existence of $G$-designs with the additional property that they contain no proper subsystems has been previously settled when $G\in \{K_3,K_4–e\}$. In this paper, the existence of $P_m$-designs which contain no proper subsystems is completely settled for every value of $m$ and $v$.