Giovanni Lo Faro

The Doyen-Wilson theorem for kite systems

Necessary and sufficient conditions are given to embed a kite system of order n into a kite system of order m.

Intersection Numbers of Kirkman Triple Systems

LetJR(v) denote the set of all integersksuch that there exists a pair ofKTS(v) with preciselyktriples in common. In this article we determine the setJR(v) forv?3(mod6) (only 10 cases are left undecided forv=15, 21, 27, 33, 39) and establish thatJR(v)=I(v) forv?3(mod6) andv?45, whereI(v)={0, 1, ?, tv?6, tv?4, tv} andtv=16v(v?1).

The flower intersection problem for Kirkman triple systems

The flower at a point x in a Steiner triple system is the set of all triples containing x. Denote by the set of all integers k such that there exists a pair of KTS(2r+1) having k+r triples in common, r of them being the triples of a common flower. In this article we determine the set for any positive integer (only nine cases are left undecided for r=7,13,16,19), and establish that for and r⩾22 where J[r]={0,1,…,2r(r−1)/3−6,2r(r−1)/3−4,2r(r−1)/3}.

On the n *-complete hypergroups

Abstract The class of n ∗ -complete hypergroups is introduced. Several properties and examples are found and a geometric interpretation is given by means of hypergraphs.

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.

Resolvable 3-star designs

Let K v be the complete graph of order v and F be a set of 1-factors of K v . In this article we study the existence of a resolvable decomposition of K v - F into 3-stars when F has the minimum number of 1-factors. We completely solve the case in which F has the minimum number of 1-factors, with the possible exception of v ? { 40 , 44 , 52 , 76 , 92 , 100 , 280 , 284 , 328 , 332 , 428 , 472 , 476 , 572 } .

Isomorphism classes of the hypergroups of type U on the right of size five

By means of a blend of theoretical arguments and computer algebra techniques, we prove that the number of isomorphism classes of hypergroups of type U on the right of order five, having a scalar (bilateral) identity, is 14751. In this way, we complete the classification of hypergroups of type U on the right of order five, started in our preceding papers [M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five, Far East J. Math. Sci. 26(2) (2007) 393-418; M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five Part two, Mathematicki Vesnik 60 (2008) 23-45; M. De Salvo, D. Freni, G. Lo Faro, On the hypergroups of type U on the right of size five, with scalar identity (submitted for publication)]. In particular, we obtain that the number of isomorphism classes of such hypergroups is 14865.

The fine triangle intersection problem for ( K 4 - e ) -designs

Let F i n ( v ) = { ( s , t ) : ? ?a?pair?of? ( K 4 - e ) -designs?of?order? v ?intersecting?in? s ?blocks?and? 2 s + t ?triangles } . Let A d m ( v ) = { ( s , t ) : s + t ? b v , s ? J ( v ) , 2 s + t ? J T ( v ) } ? { ( b v - 3 , 1 ) } , where J ( v ) (or J T ( v ) ) denotes the set of positive integers s (or t ) such that there exists a pair of ( K 4 - e ) -designs of order v intersecting in s blocks (or t triangles), and b v = v ( v - 1 ) / 10 . It is established that F i n ( v ) = A d m ( v ) for any integer v ? 0 , 1 ( mod 5 ) , v ? 6 and v ? 10 , 11 .

Partial Parallel Classes in Steiner System S(2, 3, 19)

Abstract In this paper we prove that 5 is the largest number such that every 5(2, 3, 19) has a partial parallel class of size 5.

Uniformly resolvable decompositions of K v into paths on two, three and four vertices

In this paper we consider uniformly resolvable decompositions of the complete graph K v , i.e.,?decompositions of K v whose blocks can be partitioned into factors and each factor contains pairwise isomorphic blocks. We determine necessary and sufficient conditions for the existence of a uniformly resolvable decomposition of K v into paths on two, three and four vertices.