Block colourings of 6-cycle systems

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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 \(c\).