Equitable block colourings

Let Σ = (X, B) a 4-cycle system of order v = 1 + 8k. A c-colouring of type s is a map φ∶ B → 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 ≥ 0. φ is 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 2 c ∈ {s, s + 1, . . . , ⌊ 2s 3 +s ⌋}.