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 \(v\).