Triangulating remnants of complete graphs (Q1087891)

Let U be a subgraph of \(K_ n\) that is the vertex-disjoint union of cycles and let R be the 'remnant' \(K_ n\setminus U\). This note is concerned with the conjecture that if \(| E(R)| \equiv 0 (mod 3)\) then R can be decomposed into edge-disjoint triangles. It is related to the existence, for n odd, of so-called totally symmetric quasigroups. An example is given \((n=9\) and U is the vertex-disjoint union of \(C_ 4\) and \(C_ 5)\) when the conjecture fails but the possibility that it is true for \(n>9\) remains.