Block colourings of 6-cycle systems

Abstract

Let $\Sigma=(X,\mathcal{B})$ be a $6$-cycle system of order $v$, so $v\equiv 1,9\mod 12$. A $c$-colouring of type $s$ is a map $\phi\colon\mathcal {B}\rightarrow \mathcal{C}$, with $C$ set of colours, such that exactly $c$ colours are used and for every vertex $x$ all the blocks containing $x$ are coloured exactly with s colours. Let $\frac{v-1}{2}=qs+r$, with $q, r\geq 0$. $\phi$ is equitable if for every vertex x the set of the $\frac{v-1}{2}$ blocks containing $x$ is partitioned in $r$ colour classes of cardinality $q+1$ and $s-r$ colour classes of cardinality $q$. In this paper we study bicolourings and tricolourings, for which, respectively, $s=2$ and $s=3$, distinguishing the cases $v=12k+1$ and $v=12k+9$. In particular, we settle completely the case of $s=2$, while for $s=3$ we determine upper and lower bounds for $c$.