Equitable block colourings

Let $Σ=(X,\mathcal B)$ a $4$-cycle system of order $v=1+8k$. A $c$-colouring of type $s$ is a map $ϕ\colon \mathcal B\rightarrow \mathcal C$, 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 $4k=qs+r$, with $q,r\ge 0$. $ϕ$ is \emph{equitable} if for every vertex $x$ the set of the $4k$ blocks containing $x$ is parted in $r$ colour classes of cardinality $q+1$ and $s-r$ colour classes of cardinality $q$. In this paper we study colourings for which $s|k$, giving a description of equitable block colourings for $c\in \{s,s+1,\dots,\lfloor\tfrac{2s^2+s}{3}\rfloor \}$.

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