The Nonexistence of a Decomposition of Complete Graph $K_{1}2$ into Three Factors with Diameter Two

Abstract

In this paper we show that the complete graph $K_{12}$
is not decomposable into three factors of diameter two, thus
resolving a longstanding open problem. This result completes
the solution of decomposition of a complete graph into three
factors, one of which has diameter two and the other factors
have finite diameters.