Block colourings of 6-cycle systems

LetΣ=(X,B)bea6-cyclesystemoforderv,sov≡1,9 mod12.Ac-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 v−1 = qs + r, with q,r ≥ 0. φ is equitable if for every vertex x the set of the v−1 blocks 22 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.