Equitable block colourings for 8-cycle systems

Let Σ = (X, B) be an 8-cycle system of order v = 1+16k. A c-colouring of type s is a map φ: B →C, with C set of colours, so that exactly c colours are used and for every vertex x all the blocks containing x are coloured with exactly s colours. Let 8k = qs+r, with q, r ≥ 0. The colouring φ is called equitable if for every vertex x the set of the 8k blocks containing x is partitioned into r colour classes of cardinality q +1 and s− r colour classes of cardinality q. This paper deals with a study of bicolourings, tricolourings and quadricolourings with s = 2, 3, 4.