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

A block-colouring of a 4-cycle system (V,B) of order v=1+8k is a mapping ?:B→C, where C is a set of colours. Every vertex of a 4-cycle system of order v=8k+1 is contained in 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 ith 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.