Some 2-Coloured 5-Cycle Decompositions

Let \( C \) be the set of distinct ways in which the vertices of a \( 5 \)-cycle may be coloured with at most two colours, called \emph{colouring types}, and let \( S \subseteq C \). Suppose we colour the vertices of \( K_v \) with at most two colours. If \( \mathcal{D} \) is a \( 5 \)-cycle decomposition of \( K_v \), such that the colouring type of each \( 5 \)-cycle is in \( S \), and every colouring type in \( S \) is represented in \( \mathcal{D} \), then \( \mathcal{D} \) is said to have a \emph{proper colouring type} \( S \). For all \( S \) with \( |S| \leq 2 \), we determine some necessary conditions for the existence of a \( 5 \)-cycle decomposition of \( K_v \) with proper colouring type \( S \). In many cases, we show that these conditions are also sufficient.