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

A block-colouring of a 4-cycle system ( V , B ) of order v = 1 + 8 k is a mapping ϕ : B → C , where C is a set of colours. Every vertex of a 4-cycle system of order v = 8 k + 1 is contained in r = v − 1 2 = 4 k blocks and r is called, using the graph theoretic terminology, the degree or the repetition number. 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 , let B x , i be the set of all the blocks incident with x and coloured with the i th colour. A colouring of type s is equitable if, for every vertex x , we have | B x , i − B x , j | ≤ 1 , for all i , j = 1 , … , s . In this paper we study bicolourings, tricolourings and quadricolourings, i.e. the equitable colourings of type s with s = 2 , s = 3 and s = 4 , for 4-cycle systems.