Construction of BP3-designs with mononumerical spectrum
Abstract
A mixed hypergraph is a triple H=(X;C,D), where X is the vertex set and each of C,D is a list of nonempty subsets of X: the C-edges and the D-edges. A strict k-colouring of H is a surjection c:X→{1,2,...,k} such that each edge of C has at least two vertices assigned a common value and each edge of D has at least two vertices assigned distinct values. If for each j=1,2,...,n , rj is the number of partitions of X into j nonempty parts (the colour classes) such that the colouring constraint is satisfied on each edge, then the vector R(H)=(r1, r2,..., rn) is the chromatic spectrum of H. In this paper we examine colourings of mixed hypergraphs in the case that H is a P3-design and we construct families of P3-designs having chromatic spectrum with exactly one nonzero value.
