Some 2-Coloured 5-Cycle Decompositions

Abstract

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 $|S|\le 2$, 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.