Voloshin's colourings of P3-designs

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, called the C-edges and the D-edges, respectively. A proper k-colouring of H is a mapping f:X→{1,2,…,k} such that each C-edge has at least two vertices of a common colour and each D-edge has at least two vertices of distinct colours. If rj is the number of partitions of X into j colour classes such that the colouring constraint is satisfied on each C- and each D-edge, then the vector R(H)=(r1,r2,…,rn) is the chromatic spectrum of H. Chromatic spectrum is broken if there exist integers i<j<k such that ri>0 and rk>0 but rj=0. Mixed hypergraph is uncolourable if it admits no proper colouring. The maximum (minimum) i such that ri≠0 is the upper (lower) chromatic number of a mixed hypergraph H denoted by χ̄(H)(χ(H)). In this paper we examine colourings of mixed hypergraphs in the case that H is a P3-design, i.e. a P3-decomposition of the complete graph Kn, and construct families of uncolourable P3-designs, P3-designs having the chromatic spectrum broken, P3-designs having lower and upper chromatic number equal, P3-designs colourable only by χ and χ̄ colours.