Some Problems About \(R\)-Factorizations of Complete Graphs

We propose a number of problems about \(r\)-factorizations of complete graphs. By a completely novel method, we show that \(K_{2n+1}\) has a \(2\)-factorization in which all \(2\)-factors are non-isomorphic. We also consider \(r\)-factorizations of \(K_{rn+1}\) where \(r \geq 3\); we show that \(K_{rn+1}\) has an \(r\)-factorization in which the \(r\)-factors are all \(r\)-connected and the number of isomorphism classes in which the \(r\)-factors lie is either \(2\) or \(3\).