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

A block-colouring of a 4-cycle system (V,B)(V,B) of order v=1+8kv=1+8k is a mapping ϕ:B→Cϕ:B→C, where CC is a set of colours. Every vertex of a 4-cycle system of order v=8k+1v=8k+1 is contained in r=v−12=4k blocks and rr is called, using the graph theoretic terminology, the degree or the repetition number  . A partition of degree rr into ss parts defines a colouring of type ss in which the blocks containing a vertex xx are coloured exactly with ss colours. For a vertex xx and for i=1,2,…,si=1,2,…,s, let Bx,iBx,i be the set of all the blocks incident with xx and coloured with the iith colour. A colouring of type ss is equitable   if, for every vertex xx, we have |Bx,i−Bx,j|≤1|Bx,i−Bx,j|≤1, for all i,j=1,…,si,j=1,…,s. In this paper we study bicolourings, tricolourings and quadricolourings, i.e. the equitable colourings of type ss with s=2s=2, s=3s=3 and s=4s=4, for 4-cycle systems.