On upper chromatic number for SQS(10) and SQS(16)

  • Lorenzo Milazzo


A mixed hypergraph is characterized by the fact that it possesses anti-edges as well as edges. In a colouring of a mixed hypergraph, every anti-edge has at least two vertices of the same colour and every edge has at least two vertices coloured differently. The upper chromatic number X is the maximal number of colours for which there exists a colouring using all the colours. The concepts of mixed hypergraph and upper chromatic number are applied to STS and SQS. In fact it is possible to consider a Steiner system as a mixed hypergraph when all the blocks are anti-edges (Co-STSs, Co-SQSs) or at the same time edges and anti-edges (BSTSs, BSQSs). In this paper the necessary conditions in order to colour Co-STSs, BSTSs and Co-SQSs, BSQSs are given and the values of upper chromatic number for Co-SQS(10), BSQS(10) and for BSQSs(16), obtained from a doubling construction, are determined.