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

A block colouring of a C(4)- system (V,B) of order v=1+8k is a mapping ϕ:B->C, where C is a set of colours. Every vertex of a C(4)-system of order v=8k+1 is contained in r=(v-1)/2=4k blocks and r is called 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, it is |B(x,i)-B(x,j)|<=1, for all i,j=1,...,s. In this paper the authors study equibicolourings, equitricolourings and equiquadricolourings, i.e. equitable colourings of type s with s=2, s=3, s=4, for C(4)-systems.

In questo lavoro gli autori studiano colorazioni dei blocchi di C(4)-systems che siano equipartite in s parti. Si prendono in esame i casi in cui s=2, s=3, s=4 con la determinazione del numero cromatico "upper" per i C(4)-systems.