Mario Gionfriddo

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.

On the existence of uniformly resolvable decompositions of Kv and Kv−I into paths and kites

Abstract In this paper, it is shown that, for every v ≡ 0 ( mod 12 ) , there exists a uniformly resolvable decomposition of K v - I , the complete undirected graph minus a 1 -factor, into r classes containing only copies of 2-stars and s classes containing only copies of kites if and only if ( r , s ) ∈ { ( 3 x , 1 + v − 4 2 − 2 x ) , x = 0 , … , v − 4 4 } . It is also shown that a uniformly resolvable decomposition of K v into r classes containing only copies of 2-stars and s classes containing only copies of kites exists if and only if v ≡ 9 ( mod 12 ) and s = 0 .

On conjectures of Berge and Chva´tal

Abstract We investigate the relations among the chromatic index q( H ), the maximum degree Δ( H ), the total chromatic number q∗( H ), and the maximum size Δ0( H ) of an intersecting subhypergraph of a hypergraph H . For some particular classes of hypergraphs, including Steiner systems, we provide sufficient conditions insuring that some (or all) of the trivial inequalities Δ( H )⩽Δ0( H )⩽q( H )⩽q∗( H turn to equality. For instance, we prove that Δ( H )= Δ 0 ( H ) holds whenever the maximum degree of a hypergraph H is sufficiently large with respect to the rank and the ‘pair-degree’ of H .

2-Colourings in S(t, t + 1, v)

Let S(t, k, v) be any nontrivial Steiner system. In this paper we prove the nonexistence of 2-colourings in Steiner systems S(t, t + 1, v) when t + 1 is an odd number. Further, we prove that if t + 1 is an even number and C is a blocking set of the system S(t, t + 1, v) then ¦C¦=v/2.

Octagon kite systems

Abstract The spectrum of octagon kite system (OKS) which is nesting strongly balanced 4-kite-designs is determined.

Blocking sets in 3-designs

We characterize the existence of blocking sets in a particular class of Sλ (3,4,v) and we give a construction to obtain Sλ(3,4,2v) having blocking sets.

Some Results on Partial Steiner Quadruple Systems

Abstract In this paper we prove some results concerning DMB PQSs having at least an element of degree seven and with m = 14 or 15 blocks. In a previous paper we have determined, to within isomorphism, all DMB PQSs having 12, 14, 15 blocks and satisfying some particular conditions. By means of the results contained in this paper, we purpose to construct all DMB PQSs having 12, 14, 15 blocks.

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.

A result concerning two conjectures of Berge and Chva´tal

Abstract We prove that two famous conjectures of Berge and of Chvatal are true for all nearly resolvable Steiner triples systems.