Lorenzo Milazzo

Upper chromatic number of Steiner triple and quadruple systems

Abstract The upper chromatic number χ ( H ) of a set system H is the maximum number of colours that can be assigned to the elements of the underlying set of H in such a way that each H ϵ H contains a monochromatic pair of elements. We prove that a Steiner triple system of order v ⩽ 2k − 1 has an upper chromatic number which is at most k. This bound is the best possible, and the extremal configurations attaining equality can be characterized. Some consequences for Steiner quadruple systems are also obtained.

Strict colouring for classes of Steiner triple systems

Abstract We investigate the largest number of colours, called upper chromatic number and denoted X ( H ) , that can be assigned to the vertices (points) of a Steiner triple system H in such a way that every block H ∈ H contains at least two vertices of the same colour. The exact value of X is determined for some classes of triple systems, and it is observed further that optimal colourings with the same number of colours exist also under the additional assumption that no monochromatic block occurs. Examples show, however, that the cardinalities of the colour classes in the latter case are more strictly determined.

2-Regular equicolourings for P 4 -designs

Abstract Let G be a graph. Then a G -decomposition of K v , a complete graph on v vertices, is a pair Σ = ( X , B ) , where X is the vertex set of K v and B is a partition of the edge set of K v into graphs all isomorphic to G . The elements of B are called blocks and Σ is said to be a G -design of order v . In this paper we study colourings of P 4 -designs where, in each block of B , two vertices are assigned the same colour and the other two another colour. We determine, among other things, families of P 4 -designs having a chromatic spectrum with gaps. These are the only known cases of G -designs having this property except for the families of P 3 -designs found by Lucia Gionfriddo.

Bicolouring Steiner systems S(2,4,v)

We discuss colourings of elements of Steiner systems S(2,4,v) in which the elements of each block get precisely two colours. We show that there exist systems admitting such colourings with arbitrary large number of colours, as well as systems which are uncolourable.

Enumerating and decoding perfect linear Lee codes

Using group theory approach, we determine all numbers q for which there exists a linear 1-error correcting perfect Lee code of block length n over Zq, and then we enumerate those codes. At the same time this approach allows us to design a linear time decoding algorithm.

The monochromatic block number

In 1993 Voloshin introduced the concept of mixed hypergraph. A mixed hypergraph is characterised 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. In this paper a new parameter is introduced: the monochromatic block number mb. It is the number of monochromatic blocks in a colouring of a Steiner system. The exact value of mb is given for every colouring in STS(v). Upper and lower bounds for mb in particular SQSs are found.

Strict colorings of Steiner triple and quadruple systems: a survey

The paper surveys problems, results and methods concerning the coloring of Steiner triple and quadruple systems viewed as mixed hypergraphs. In this setting, two types of conditions are considered: each block of the Steiner system in question has to contain (i) a monochromatic pair of vertices, or, more, restrictively, (ii) a triple of vertices that meets precisely two color classes.

Equitable Specialized Block-Colourings for Steiner Triple Systems

We continue the study of specialized block-colourings of Steiner triple systems initiated in [2] in which the triples through any element are coloured according to a given partition π of the replication number. Such colourings are equitable if π is an equitable partition (i.e., the difference between any two parts of π is at most one). Our main results deal with colourings according to equitable partitions into two, and three parts, respectively.

Extended Bicolorings of Steiner Triple Systems of Order

A bicoloring of a Steiner triple system STS(n) on n vertices is a coloring of vertices in such a way that every block receives precisely two colors. The maximum (resp. minimum) number of colors in a bicoloring of an STS(n) is denoted by χ (resp. χ). All bicolorable STS(2 − 1)s have upper chromatic number χ ≤ h; also, if χ = h < 10, then lower and upper chromatic numbers coincide, namely, χ = χ = h. In 2003, M. Gionfriddo conjectured that this equality holds whenever χ = h ≥ 2. In this paper we discuss some extensions of bicolorings of STS(v) to bicoloring of STS(2v+ 1) obtained by using the ‘doubling plus one construction’. We prove several necessary conditions for bicolorings of STS(2v+1) provided that no new color is used. In addition, for any natural number h we determine a triple system STS(2 − 1) which admits no extended bicolorings.

2^{h}-1

A partial directed 2k-bicycle system of order (s,t) can always be embedded in a directed 2k-bicycle system of order (ks,kt).