Embedding coverings of 2-paths with 3-paths (Q1876693)

A {\((3,2)\)-path covering} of a complete graph \(K_n\) is a set \(S\) of \(3\)-paths of \(K_n\) such that each, except possibly one, \(2\)-path of \(K_n\) is contained in a \(3\)-path from \(S\). The paper solves the embedding problem for \((3,2)\)-path coverings of complete graphs. It is proved that any \((3,2)\)-path covering of \(K_n\) can be embedded in a \((3,2)\)-path covering of \(K_{n+m}\) if and only if the following holds: if \(m=1\) then \(n=1\), and if \(n=1\) then \(m\neq 2\).