giovanni Lo Faro - Profile on Academia.edu (original) (raw)

Papers by giovanni Lo Faro

Mathematics, Mar 15, 2022

Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose co... more Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose components are isomorphic to H. Given a set H of mutually non-isomorphic graphs, a uniform H-factorization of G is a partition of the edges of G into H-factors for some H ∈ H. In this article, we give a complete solution to the existence problem for uniform (C k , P k+1 )-factorizations of K n -I in the case when k is even.

Australas. J Comb., 1995

An extended triple system with no idempotent element (ETS) is a collection of non ordered triples... more An extended triple system with no idempotent element (ETS) is a collection of non ordered triples of type {x,y,z} or {x,x,y} chosen from a v -set in such a way that each pair (whether distinct or not) is contained in exactly one triple. (For example, in the block {x,x,y}, the pair {x,y} is said to occur one tinle.) Such a design has "'v = v(v + 3) /6 blocks and a necessary and sufficient condition for existence is that v ° (mod 3). Let J( v) denote the set of non -negative integers k such that there exist two ETS(v) with precisely k blocks in common. In this paper we determine J( v) for all admissible v, in particular we show that J( ) 1( )-{13} and J(v) =1(v), where l{v) ={O,l, ... , sv-3, s~} The concept of an extended triple system was introduced by D.M. Johnson and N.S. Mendelsohn . An extended triple system is a pair (V,B), where V is a finite set and B is a collection of non -ordered triples from V , where each triple may have repeated elements, such that every pair of elements of V, not necessarily distinct, is contained in exactly one triple of B. The triple of B are of three types (1) {x,x,:r} , (2) {y,y,z} and (3) {a,b,c} , where the element x is called an idempotent and y a non -idempotent of the system (V,B). We shall denote by {v ; a} the class of all extended triple systems on v -elements containing exactly 0: idempotent elements.

Australas. J Comb., 2002

Let J * (v) be the set of all integers k such that there is a pair of Latin squares L and L with ... more Let J * (v) be the set of all integers k such that there is a pair of Latin squares L and L with their own orthogonal mates on the same v-set, and with L and L having k cells in common. In this article we completely determine the set J * (v) for integers v ≥ 24 and v = 1, 3, 4, 5, 8, 9. For v = 7 and 10 ≤ v ≤ 23, there are only a few cases left undecided for the set J * (v).

Journal of the Australian Mathematical Society, Oct 1, 1994

It has been conjectured that for any union-closed set S& there exists some element which is conta... more It has been conjectured that for any union-closed set S& there exists some element which is contained in at least half the sets in s/. It is shown that this conjecture is true if the number of sets in &/ is less than 25. Several conditions on a counterexample are also obtained, 1991 Mathematics subject classification (Amer. Math. Soc): 05A05,05A99.

The Australasian Journal of Combinatorics, Dec 1, 1998

An extended triple system (a twofold extended triple system) with no idempotent element (ETS, TET... more An extended triple system (a twofold extended triple system) with no idempotent element (ETS, TETS respectively) is a pair (V, B) where V is a v-set and B is a collection of unordered triples, called blocks, of type {x,y,z} or {x,x,y}, such that each pair (whether distinct or not) is contained in exactly one (respectively, exactly two) blocks. For example, in the block {x, x, y}, the occurrence of the pair {x, y} is counted once. It is well-known that an ETS( v) of order v (ETS( v)) exists if and only if v == 0 (mod 3), and it is trivial to see that a TETS of order v (TETS(v)) exists if and only if v == 0 (mod 3). If a TETS (v) contains two blocks b l , b 2 that are identical as subsets of V, then b i = b 2 is said to be a repeated block. We are interested in the following question: Given v == 0 (mod 3) and a nonnegative integer k, does there exist a TETS(v) with exactly k repeated blocks? This question is related to the intersection problem for ETSs, solved by Lo Faro in 1995. The same question with the additional condition that the TETS be indecomposable (that is, cannot have its blocks partitioned into two ETS) is also of interest. The purpose of this paper is to completely settle these questions.

arXiv (Cornell University), Mar 5, 2015

In this article we completely determine the spectrum for uniformly resolvable decompositions of t... more In this article we completely determine the spectrum for uniformly resolvable decompositions of the complete graph K v into r 1-factors and s classes containing only copies of h-suns.

Discrete Mathematics, Apr 1, 2015

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 ... more 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 1factors, with the possible exception of v ∈ {40,

Discrete Mathematics, Feb 1, 1993

Let S(t, k, c) be any nontrivial Steiner system. In this paper we prove the nonexistence of 2colo... more Let S(t, k, c) be any nontrivial Steiner system. In this paper we prove the nonexistence of 2colourings in Steiner systems S(t, t + 1, ~1) 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, L') then ICI =n/2. Very little is known today about the existence of blocking sets in Steiner systems. Results on blocking sets in Steiner quadruple systems have been obtained by Doyen

Discrete Mathematics, Dec 1, 2015

In this paper we consider uniformly resolvable decompositions of the complete graph K v into subg... more In this paper we consider uniformly resolvable decompositions of the complete graph K v into subgraphs such that each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either P 2 , P 3 and P 4 .

$Research paper thumbnail of Resolvable G-designs of order v and index {\lambda}$

arXiv (Cornell University), Mar 2, 2015

In this paper we consider the problem concerning the existence of a resolvable G-design of order ... more In this paper we consider the problem concerning the existence of a resolvable G-design of order v and index λ. We solve the problem for the cases in which G is a connected subgraph of K 4 .

arXiv (Cornell University), Feb 27, 2014

It is established that up to isomorphism, there are only one (K 4e)design of order 6, three (K 4e... more It is established that up to isomorphism, there are only one (K 4e)design of order 6, three (K 4e)-designs of order 10 and two (K 4e)-designs of order 11. As an application of our enumerative results, we discuss the fine triangle intersection problem for (K 4e)-designs of orders v = 6, 10, 11.

Australas. J Comb., 2000

In this paper the first BSTS in which the upper chromatic number is different from the lower chro... more In this paper the first BSTS in which the upper chromatic number is different from the lower chromatic number is determined. It is a BSTS( ) and its order is the lowest for which this property holds. In addition, all the possible strict colourings and their upper and lower chromatic number for systems of triples of the type BSTS(15), BSTS(19) and BSTS( ) are also determined.

Three types of edge-switchable kite systems

Ars Combinatoria, 2011

ABSTRACT Informally, an ϵ-switchable G-design is a decomposition of the complete graph into subgr... more ABSTRACT Informally, an ϵ-switchable G-design is a decomposition of the complete graph into subgraphs of isomorphic copies of G which have the property that they remain a G-decomposition when ϵ-edge switches are made to the subgraphs. This paper determines the spectrum of ϵ-switchable G-design where G is a kite (a triangle with an edge attached) and ϵ takes t-edge, h-edge and l-edge.

On Block Sharing Steiner Quadruple Systems

Elsevier eBooks, 1986

Annals of Discrete Mathematics 30 (1986) 297-302 © Elsevier Science Publishers BV (North-Holland)... more Annals of Discrete Mathematics 30 (1986) 297-302 © Elsevier Science Publishers BV (North-Holland) 297 ON BLOCK SHARING STEINER QUADRUPLE SYSTEMS Giovanni LO FARO (*) Dipartimento di Matematica del 1'Universita, Via C. Battisti 90, 98100 MESSINA, Italy ...

Ars Mathematica Contemporanea, Sep 20, 2018

A solution to the existence problem of G-designs with given subdesigns is known when G is a trian... more A solution to the existence problem of G-designs with given subdesigns is known when G is a triangle with p = 0, 1, or 2 disjoint pendent edges: for p = 0, it is due to Doyen and Wilson, the first to pose such a problem for Steiner triple systems; for p = 1 and p = 2, the corresponding designs are kite systems and bull designs, respectively. Here, a complete solution to the problem is given in the remaining case where G is a 3-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor.

Discrete Mathematics, Sep 1, 2005

Strict colourings of STS(3v)s containing three mutually disjoint subsystems of order v are examin... more Strict colourings of STS(3v)s containing three mutually disjoint subsystems of order v are examined. An upper bound for ¯ of STS(3v) linked with ¯ of its subsystems of order v is determined and an infinite class of uncolourable BSTS(3v)s is found.

The intersection problem for twin bowtie and near bowtie systems

Ars Combinatoria, 2001

ABSTRACT The authors completely determine the set of numbers k such that there exists a pair of t... more ABSTRACT The authors completely determine the set of numbers k such that there exists a pair of twin bowtie systems (or twin near bowtie systems) intersecting in k bowties. The proof is constructive. A bowtie in a graph is a pair of triangles having exactly one vertex in common. A bowtie system is a set of edge disjoint bowties partitioning the edge set of a complete graph.

Discrete Mathematics, 2009

Let (X, B) be a (λK v , G 1 )-design and G 2 a subgraph of G 1 . Define sets B(G 2 ) and D(G 1 \ ... more Let (X, B) be a (λK v , G 1 )-design and G 2 a subgraph of G 1 . Define sets B(G 2 ) and D(G 1 \ G 2 ) as follows: for each block B ∈ B, partition B into copies of G 2 and G 1 \ G 2 and place the copy of G 2 in B(G 2 ) and the edges belonging to the copy of ) denote the set of all integers v such that there exists a (λK v , G 1 > G 2 )-design. In this paper we completely determine the set Meta(K 3 + e > P 4 , λ) or Meta(K 3 + e > H 4 , λ) when the admissible conditions are satisfied, for any λ.