Block colourings of 6-cycle systems

Abstract

Abstract. Let Σ = (X, B) be a 6-cycle system of order v, so v ≡ 1, 9 mod 12. 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= qs + r, with q, r ≥ 0. φ is equitable if for every vertex x the set of the 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.