Some $2$-Coloured $4$-Cycle Decompositions

Abstract

There are six distinct ways in which the vertices of a 4-cycle may be coloured with two colours, called $\text{colouring types}$. Let $C$ be the set of these colouring types and let $S$ be a non-empty subset of $C$. Suppose we colour the vertices of $K_v$ with two colours. If $D$ is a 4-cycle decomposition of $K_v$ such that the colouring type of each 4-cycle is in $S$, then $D$ is said to have a $colouringoftype$ $S$. Furthermore, the colouring is said to be $proper$ if every colouring type in $S$ is represented in $D$. For all possible $S$ of size one, two or three, excluding three cases already settled, we completely settle the existence question for 4-cycle decompositions of $K_v$ with a colouring of type $S$.