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Block colourings of 6-cycle systems

Bonacini P., Gionfriddo M., Marino L.

Opuscula Mathematica

2017

Vol. 37, no. 5

647--664

Let Σ = (X, Β) be a 6-cycle system of order v, so v ≡ 1, 9 mod 12. A c-colouring of type s is a map φ: Β) → 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 [formula], with q,r ≥ 0. φ is equitable if for every vertex x the set of the [formula] 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.