Equitable specialized block-colourings for 4-cycle systems — II

A block-colouring of a 4 -cycle system ( X , B ) of order v = 1 + 8 k is a mapping ϕ : B → Δ , where Δ is a set of colours. A partition of degree r into s parts defines a colouring of type s in which the blocks containing a vertex x are coloured exactly with s colours. For a vertex x and for i = 1 , 2 , … , s , B x , i is the set of all the blocks incident with x and coloured with colour i . A colouring of type s is equitable if, for every vertex x , | B x , i − B x , j | ≤ 1 , for all i , j = 1 , … , s . s paper the authors continue the research begun in Gionfriddo et al. (2010) [2], where in particular they had studied tricolourings with three colours. Here the authors study tricolourings, i.e. equitable colourings of type 3 , for 4 -cycle systems and, in particular, they give complete results for C 4 ( 9 ) -systems and for C 4 ( 1 + 24 h ) -systems, by proving that χ 3 ′ ¯ ( 1 + 24 h ) = 7 , and for the tricolourings with four colours.