Non-existence of (3,2)-Equicolourings in C k -Designs

Abstract A block colouring of a C k -design Σ = (X, B) of order v (odd) is a mapping φ : B → C, where blocks and d(x) is called, using graph theoretic terminology, the degree of the vertex x. A partition of degree D 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, indicate by B x,i the set of all the blocks incident in x and coloured with the colour C i . 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. If |C| = r, such a colouring will said an (r, s)-equiblock-colouring. In this paper we prove the non-existence of (r, s)-equiblock-colourings, having s = 2 and r = 3, for some classes of C 4 -designs. Mathematics Subject Classification: 05B0

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices