Some Problems About $R$-Factorizations of Complete Graphs

Abstract

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 $r\ge 3$; 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 $3$.