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Gloria RINALDI

Professore Ordinario
Dipartimento di Scienze e Metodi dell'Ingegneria

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Pubblicazioni

2023 - Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees [Articolo su rivista]
Rinaldi, Gloria
abstract

A 1-factorization F of a complete graph K2n is said to be G-regular, or regular under G, if G is an automorphism group of F acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2 was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2 is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2.

2021 - Regular 1-factorizations of complete graphs and decompositions into pairwise isomorphic rainbow spanning trees [Articolo su rivista]
Rinaldi, Gloria
abstract

A 1-factorization F of a complete graph K2n is said to be G-regular, or regular under G, if G is an automorphism group of F acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (1985) on cyclic groups, and it is still open even though several other classes of groups were tested in the recent past. An attempt to obtain a fairly precise description of groups and 1-factorizations satisfying this symmetry constraint can be done by imposing further conditions. In this paper we prove that, regardless of the isomorphism type of G, the existence of a G-regular 1-factorization of K2n together with a complete set of isomorphic rainbow spanning trees which are in the orbit of a single one is assured if and only if n ≥ 3 is an odd number. Also, when n is even, we examine dihedral groups: for each dihedral group G of order 2n ≥ 6, it is possible to exhibit a G-regular 1-factorization of K2n together with two non isomorphic rainbow spanning trees whose partial orbits give rise to a complete set. This extends some recent results obtained by Caughman et al. (2017) and by Mazzuoccolo et al. (2019) for the class of cyclic groups.

2021 - The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class [Articolo su rivista]
Bonvicini, S.; Buratti, M.; Garonzi, M.; Rinaldi, G.; Traetta, T.
abstract

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever v≡ 39 (mod 72), or v≡ 4 e48 + 3 (mod 4 e96) and e≥ 0 , there exists a KTS on v points having at least v- 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

2019 - Rainbow spanning tree decompositions in complete graphs colored by cyclic 1-factorizations [Articolo su rivista]
Rinaldi, G.; Mazzuoccolo, G.
abstract

Brualdi and Hollingsworth conjectured in Brualdi and Hollingsworth (1996) that in any complete graph K2n, n≥3, which is properly colored with 2n−1 colors, the edge set can be partitioned into n edge disjoint rainbow spanning trees (where a graph is said to be rainbow if its edges have distinct colors). Constantine (2002) strengthened this conjecture asking the rainbow spanning trees to be pairwise isomorphic. He also showed an example satisfying his conjecture for every 2n∈{2s:s≥3}∪{5⋅2s,s≥1}. Caughmann, Krussel and Mahoney (2017) recently showed a first infinite family of edge colorings for which the conjecture of Brualdi and Hollingsworth can be verified. In the present paper, we extend this result to all edge-colorings arising from cyclic 1-factorizations of K2n constructed by Hartman and Rosa (1985). Finally, we remark that our constructions permit to extend Constatine's result also to all 2n∈{2sd:s≥1,d>3odd}.

2018 - Indecomposable 1-factorizations of the complete multigraph λK2n for every λ≤2n [Articolo su rivista]
Rinaldi, Gloria; Bonvicini, Simona
abstract

A 1-factorization of the complete multigraph λK2n is said to be indecomposable if it cannot be represented as the union of 1-factorizations of λ0K2n and (λ - λ0)K2n, where λ0 < λ. It is said to be simple if no 1-factor is repeated. For every n ≥ 9 and for every (n - 2)/3 ≤ λ ≤ 2n, we construct an indecomposable 1-factorization of λK2n, which is not simple. These 1-factorizations provide simple and indecomposable 1-factorizations of λK2s for every s ≥ 18 and 2 ≤ λ ≤ 2└s/2┘ - 1. We also give a generalization of a result by Colbourn et al., which provides a simple and indecomposable 1-factorization of λK2n, where 2n = pm + 1, λ = (pm - 1)/2, p prime.

2017 - 3-pyramidal Steiner triple systems [Articolo su rivista]
Buratti, Marco; Rinaldi, Gloria; Traetta, Tommaso
abstract

A design is said to be f-pyramidal when it admits an automorphism group fixing f points and acting sharply transitiveky on all the others. The problem of establishing the set of values of v for which there exists a f-pyramidal Steiner system of order v was deeply investigated in the case f=1 but it remains open for special classes of v. The same problem for the next class of f, which is f=3, is completly solved here. There exists a 3-pyramidal Steiner triple system of order v if and only if v=7,9,15 (mod 24) or v=3,19 (mod 48).

2017 - Vertex-regular 1-factorizations of the complete graph [Articolo su rivista]
Rinaldi, Gloria
abstract

A 1-factorization of a complete graph is said to be regular if it admits an automorphism group with a sharply transitive action on the vertex set. Which abstract groups can realize such a situation? The complete answer is still unknown but the problem has been solved in some cases. We illustrate the state of art.

2014 - On 2-pyramidal Hamiltonian cycle systems [Articolo su rivista]
R., Bailey; M., Buratti; Rinaldi, Gloria; T., Traetta
abstract

A Hamiltonian cycle system of the complete graph on 2v vertices minus a 1 factor (briefly, an HCS(2v)) is 2-pyramidal if it admits an automorphism group of order 2v - 2 fixing two vertices. In spite of the fact that the very first example of an HCS(2v) is very old and 2-pyramidal, a thorough investigation of this class of HCSs is lacking. We give first evidence that there is a strong relationship between 2-pyramidal HCS(2v) and 1-rotational Hamiltonian cycle systems of the complete graph on 2v-1 vertices. Then, as main result, we determine the full automorphism group of every 2-pyramidal HCS(2v). This allows us to obtain an exponential lower bound on the number of non-isomorphic 2-pyramidal HCS (2v).

2014 - Some Results on 1-Rotational Hamiltonian Cycle Systems [Articolo su rivista]
Buratti, Marco; Rinaldi, Gloria; Traetta, Tommaso
abstract

A Hamiltonian cycle system of the complete graph on v vertices (briefly, a HCS(v)) is 1-rotational under a (necessarily binary) group G if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any integer n greatest or equal to 3, there exists a 3-perfect 1-rotational HCS(2n+1). This allows to get the existence of an infinite class of 3-perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full automorphism group of a 1-rotational HCS under G is G itself unless the HCS is the 2-transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion, a formula by Bailey, Ollis, and Preece on the number of inequivalent 1-rotational HCSs under G. This leads us to a formula counting all 1-rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, an imprtant lower bound for the number of non isomorphic 1-rotational (and hence symmetric) HCSs is obtained.

2013 - A collection of results on Hamiltonian cycle systems with a nice automorphism group [Articolo su rivista]
M., Buratti; S., Capparelli; F., Merola; Rinaldi, Gloria; T., Traetta
abstract

Some old and new results on Hamiltonian cycle systems of the complete graph (or of the complete graph minus a 1-factor) having an automorphism group that satisfies specific properties are collected.

2013 - A hierarchy of balanced graph-designs [Articolo su rivista]
Bonisoli, Arrigo; Bonvicini, Simona; Rinaldi, Gloria
abstract

Decompositions of the complete graph K_v into subgraphs, all of which are isomorphic to some given non-regular graph G are considered. The decompositions are required to have the additional property that each vertex occurs a constant number of times as a vertex of given degree in the subgraphs of the decomposition. These decompositions are said to be degree-balanced G-designs. General properties of degree-balanced G-designs are studied and the spectrum of degree-balanced G-designs is determined when G is a bowtie. Moreover, for each v in this spectrum, there exists a bowtie design on v vertices which is not degree-balanced.

2013 - Invariant Kekulé structures in fullerene graphs [Articolo su rivista]
Mathieu, Bogaerts; Giuseppe, Mazzuoccolo; Rinaldi, Gloria
abstract

Fullerene graphs are trivalent plane graphs with only hexagonal and pentagonal faces. They are often used to model large carbon molecules. A totally symmetric Kekule structure in a fullerene graph is a set of independent edges which is fixed by each automorphism of the fullerene. Starting from the complete catalog of all fullerenes with at least ten symmetries, we establish exactly which of them have at least one totally symmetric Kekule structure.

2013 - Totally Symmetric Kekule structures in fullerene graphs with ten or more symmetries [Articolo su rivista]
M., Bogaerts; G., Mazzuoccolo; Rinaldi, Gloria
abstract

Graph Theoretic fullerenes are designed to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric kekule structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene. It was recently suggested that molecules with totally symmetric kekule structures could have physical and chemical properties. Starting from a catalog given by J.Graver, we study all graph theoretic fullerenes with at least ten symmetries and we establish exaclty which of them have a totally symmetric kekule structure.

2012 - Some progress on the existence of 1-rotational Steiner Triple Systems [Articolo su rivista]
Bonvicini, Simona; M., Buratti; Rinaldi, Gloria; T., Traetta
abstract

A Steiner Triple System of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point.The spectrum of values of v for which there exists a1-rotational STS(v) under a cyclic, an abelian, or a generalized quaternion group, has beenestablished in 1981 (phelps and Rosa), in 2001 (Buratti) and in 2008 (Mishima), respectively.Nevertheless, the spectrum of values of v for which there exists a1-rotational STS(v) under an arbitrary group has not been completely determined yet.This paper is a considerable step forward to the solution of this problem.In fact, we leave as uncertain cases only those for which we have v = (p^3-p)n + 1 = 1 (mod 96)with p a prime, n =1,2,3 mod 4, and the odd part of (p^3-p)n that is square-free and without prime factors congruent to 1 mod 6.

2011 - Graph products and new solutions to Oberwolfach problems [Articolo su rivista]
Rinaldi, Gloria; Traetta, Tommaso
abstract

A method to construct simple graphs starting from known ones is introduced. This method can be applied in many different situations and when applied to regular graphs and to their decompositions, a new regular graph is obtained together with a new decomposition. Using this tecnique infinitely many new solutions to the Oberolfach problem, in both the classic and equipartite case are constructed.

2010 - Sharply transitive 1-factorizations of complete multipartite graphs [Articolo su rivista]
Mazzuoccolo, Giuseppe; Rinaldi, Gloria
abstract

Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic.

2009 - A non-existence result on cyclic cycle decomposition of the cocktail party graph [Articolo su rivista]
M., Buratti; Rinaldi, Gloria
abstract

Non existence results on cyclic cycle decompoisitions of a complete graph with a 1-factor removed are given.

2009 - On 2-factorizations of the complete graph: from the k-pyramidal to the universal property [Articolo su rivista]
Bonvicini, Simona; G., Mazzuoccolo; Rinaldi, Gloria
abstract

We consider 2-factorizations of complete graphs which possess an automorphism group fixing k\ge 0 vertices and acting sharply transitively on the others. We study the structures of such factorizations and consider the cases in which the group is either abelian or dihedral in somemore details. We prove that the class of 2-factorizations of complete graphs is universal. Namely each finite group is the full automorphism group of a 2-factorization of the class.

2008 - 1-Rotational k-Factorizations of the Complete Graph and New Solutions to the Oberwolfach Problem [Articolo su rivista]
Buratti, M; Rinaldi, Gloria
abstract

We consider k-factorizations of the complete graph that are 1-rotational under an assigned group G, namely that admit G as an automorphism group acting sharply transitively on all but one vertex. After proving that the k-factors of such a factorization are pairwise isomorphic, we focus our attention to the special case k=2, a case in which we prove the involutions of G necessarily form a unique conjugacy class. We completely characterize, in particular, the 2-factorizations that are 1-rotational under a dihedral group. Finally, we get infinite new classes of prviously unknown solutions to the Oberwolfach problem via some direct and recursive constructions.

2008 - Sharply transitive decompositions of complete graphs into generalized Petersen graphs [Articolo su rivista]
Bonisoli, Arrigo; M., Buratti; Rinaldi, Gloria
abstract

A decomposition of the complete graph K_v into copies of a subgraph G is called a sharply transitive G-decomposition if it is left invariant by an automorphism group acting sharply transitively on the vertex set of K_v. For suitable values of v we construct examples of sharply transitive G-decompositions when G is either a Petersen graph, a generalized Petersen graph or a prism.

2007 - A theoretical model to predict the age of traditional balsamic vinegar [Articolo su rivista]
Giudici, Paolo; Rinaldi, Gloria
abstract

Traditional balsamic vinegar (TBV) is aged for a long time in a set of barrels arranged in decreasing scalar volume. New cooked must is added and aliquots of product are transferred from barrel to barrel every year; this procedure generates a blend of vinegars of different ages. These ages have been described by sequences of real numbers depending upon the number of year of the barrel set and volume of vinegar transferred. A theoretic study of these sequences has shown that there is a finite limit for the age of vinegar and this upper limit can be formulated through the values of both the volume of vinegar in the barrels and the transferred one. Namely, we have proved that the real ages, in any single barrel, are strictly increasing and with a finite limit as the number of years of the barrels set goes to infinity and we calculated these limits. From a practical point of view, the proposed mathematical model allowed us to define formulas able to calculate the maximum amount of TBV that each producer can sell as a limited product, in the hypothesis that the vinegar has been reached the minimal legal age required of 12 years. In addition, since the independent agency that officially states TBV authenticity has also the authority to inspect the producer's factory and therefore to know exactly how much TBV is withdrawn, it is reasonable to hypothesize that the public agency is able to found all data required to calculate the vinegar age by the proposed model.

2007 - k–Pyramidal One–Factorizations [Articolo su rivista]
Mazzuoccolo, Giuseppe; Rinaldi, Gloria
abstract

We consider one–factorizations of complete graphs which possess an automorphism group fixing k ≥ 0 vertices and acting regularly (i.e., sharply transitively) on the others. Since the cases k = 0 and k = 1 are well known in literature, we study the case k>=2 in some detail. We prove that both k and the order of the group are even and the group necessarily contains k − 1 involutions. Constructions for some classes of groups are given. In particular we extend the result of [7]: let G be an abelian group of even order and with k − 1 involutions, a one–factorization of a complete graph admitting G as an automorphism group fixing k vertices and acting regularly on the others can be constructed.

2006 - One-factorizations of complete graphs with regular automorphism groups [Articolo su rivista]
Rinaldi, Gloria
abstract

A survey on the state of art on the problem of constructing one-factorizations of complete graph which admit an automorphism group with a sharply transitive action on the vertices except ones which are assumed to be fixed.

2005 - Minkowski tangent-circle structures and key distribution patterns [Articolo su rivista]
Bonvicini, Simona; Rinaldi, Gloria
abstract

Key distribution patterns are finite incidence structures satisfying certain properties which enables them to be applied to a problem in network key distribution. Few examples of key distribution patterns are known. We present new examples of finite Minkowski tangent-circle structures and show how to construct key distribution patterns from them.

2005 - Nilpotent 1-factorizations of the complete graph [Articolo su rivista]
Rinaldi, Gloria
abstract

For which groups G of even order 2n does a 1-factorization of the complete graph on 2n veritces exist with the property of admitting G as a sharply vertex-transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in [M. Buratti "Abelian 1-factorizations of the complete graph" Europ. J Comb. 2001, pp.291-295], we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian Sylow 2-subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1-factor.

2005 - On sharply vertex transitive 2-factorizations of the complete graph [Articolo su rivista]
M., Buratti; Rinaldi, Gloria
abstract

We introduce the concept of a 2-starter in a group G of odd order. We prove that any 2-factorization of the complete graph admitting G as a sharply vertex transitive automorphism group is equivalent to a suitable 2-starter in G. Some classes of 2-starters are studied, with special attention given to those leading to solutions of some Oberwolfach or Hamilton-Waterloo problems.

2005 - On two-transitive parabolic ovals [Articolo su rivista]
Bonisoli, Arrigo; Rinaldi, Gloria
abstract

The state of knowledge on the following problem is examined. Let P be a projective plane of odd order n with an oval S and let G be a collineation group of P fixing S. Assume G fixes a point Q on S and acts 2-transitively on S - {Q}. The usual basic question is: what can be said about the plane P, the oval S and the group G?

2005 - Quaternionic starters [Articolo su rivista]
Bonisoli, Arrigo; Rinaldi, Gloria
abstract

Let m be an integer, m >= 2 and set n = 2^m. Let G be a non-cyclic group of order 2n admitting a cyclic subgroup of order n. We prove that G always admits a starter. Therefore, there exists a one - factorization of the complete graph on 2n vertices admitting G as an automorphism group acting sharply transitively on the vertex set. For an arbitrary even n > 2 we also show the existence of a starter in the dicyclic group of order 2n.

2004 - Key distribution patterns using tangent circle structures [Articolo su rivista]
Rinaldi, Gloria
abstract

The problem of key management in a communications network is of primary importance. A key distribution pattern is an incidence structure which provides a secure method of distributing keys in a large network reducing storage requirements. It is of interest to find explicit constructions for key distribution patterns. In some paper of C. O'Keefe, examples are shown using the finite circle geometries (Minkowski, Laguerre and inversive planes); in a paper of K. Quinn examples are constructed from conics in finite projective and affine planes. In this paper, we construct some examples using the finite tangent-circle structures, introduced in a paper of Quattrocchi and Rinaldi (1988) and we give a comparison of the storage requirements.

2003 - A class of complete arcs in multiply derived planes [Articolo su rivista]
Bonisoli, Arrigo; Rinaldi, Gloria
abstract

We prove that unital-derived (q^2 - q + 1)-arcs of PG(2, q^2) still yield complete arcs after multiple derivation with respect to disjoint derivation sets on a given line.

2003 - Primitive collineation groups of ovals with a fixed point [Articolo su rivista]
Bonisoli, Arrigo; Rinaldi, Gloria
abstract

We investigate collineation groups of a finite projective plane of odd order n fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which the group fixes a point off the oval is considered. We prove that it occurs in a Desarguesian plane if and only if (n + 1)/2 is an odd prime, the group lying in the normalizer of a Singer cycle of PGL(2, n) in this case. For an arbitrary plane we show that the group cannot contain Baer involutions and derive a number of structural and numerical properties in the case where the group has even order. The existence question for a non-Desarguesian example is addressed but remains unanswered, although such an example cannot have order n less than or equal to 23 as computer searches carried out with GAP show.

2002 - Complete Unital derived arcs in The Hall Plane of order 9 [Articolo su rivista]
Rinaldi, Gloria; F., Zironi
abstract

Complete 7-arcs and complete 8-arcs are constructed in the hall plane of order 9 as intersections of unitals.

2001 - Complete Unital derived arcs in the Hall Planes [Articolo su rivista]
Rinaldi, Gloria
abstract

Let q be an odd prime power. The existence of complete arcs with q^2-q+1 points in the Hall plane of order q is proved.

2001 - Finite Minkowski Planes and embedded Inversive Planes [Articolo su rivista]
Rinaldi, Gloria
abstract

It is proved that each known finite Minkowski plane of order p^m, with p prime and m even, contains embedded Miquelian inversive planes.

2001 - Inversive planes, Minkowski planes and regular sets of points [Articolo su rivista]
Rinaldi, Gloria
abstract

New examples of regular sets of points for the Miquelian inversive planes of order q, q a prime power, q greater than or equal to 7, are found and connections between such planes and certain Minkowski planes of order q(2) are presented.

2001 - Regular sets of points in finite Minkowski planes [Articolo su rivista]
Rinaldi, Gloria; F., Zironi
abstract

Let M be a Minkowski plane and let G be the automorpphism gropup of M. A set I of points of M is said to be regular if the identity is the unique automorphism of G mapping I onto itself. The set I is called a IR-set if it is a regular set of independent points. We prove that each known finite Minkoeski plane contain IR-stes except for the planes of order 4, 4 and 7 respectively. We find all the non-equivalent IR-sets contained in the known Minkowski planes of order 8 and 9 respectively.

2000 - Arcs in Laguerre Planes [Articolo su rivista]
C., Bisi; Rinaldi, Gloria
abstract

k-arcs in Laguerre planes are investigated with particular attention to problems of existence and completeness.

1999 - Transformation of projective planes and permutation sets [Articolo su rivista]
Rinaldi, Gloria
abstract

A new method for transforming incidence structures and sharply multiply transitive permutation sets was introduced in Quattrocchi and Rosati (Geom. Dedicata 44 (1992) 233-240). When applied to projective planes this method resembles Ostrom's derivation technique, (Ostrom, Trans. Amer. Math. Sec. II (1964) 1-18), but does not coincide with it. In the Section 2 we give sufficient conditions to transform a finite (infinity)- l(infinity) transitive projective plane into a plane which is still (infinity)- l(infinity) transitive. Furthermore, we apply the transformation method to the construction of flocks (i.e. sharply 1-transitive permutation sets). In the Section 3 we refine the transformation method of Quattrocchi and Rosati (Geom. Dedicata 44 (1992) 233-240) and obtain a reversible transformation procedure. (C) 1999 Elsevier Science B.V. All rights reserved.

1999 - Transformation systems for incidence structures [Articolo su rivista]
Rinaldi, Gloria; L. A., Rosati
abstract

By taking into account the transformation technique of Quattrocchi and Rosati, we study how to generate transformation systems for an incidence structure by starting from some given generating blocks and by a suitable permutation of the points. Furthermore, the method is applied to obtain the finite Andre planes by means of a minimal set of generating lines of Desarguesian planes.

1997 - Exstension of multiply transitive permutation sets [Articolo su rivista]
Rinaldi, Gloria; S., Spaggiari
abstract

The problem of extending (sharply) k-transitive permutation sets into (sharply) (k+1)-transitive permutation sets is studied. We give sufficient conditions for the extension. These conditions reduce to a unique necessary one in case the starting set is a group. Furthermore we establis some necessary and sufficient condition for a sharply k-transitive permutation sets (k>=3) to be a group.

1997 - Insiemi di permutazioni strettamente 4-transitivi e gruppo di Mathieu M_11 [Articolo su rivista]
Rinaldi, Gloria; Quattrocchi, Pasquale
abstract

Let G be a sharply 4-transitive permutation set acting on a finite set E with at least 7 points. Suppose G to contain the identity and let J be the set of involutions in G. Three conditions on J are given with forces E to contain exactly 11 points and G to be the Mathieu group M_11.

1997 - Steiner systems and n^{-1}-isomorphisms [Articolo su rivista]
Rinaldi, Gloria; G., Quattrocchi
abstract

The concept on n^{-1}-isomorphisms between Steiner Systems is introduced. These isomorphisms are used to study net replacements in affine planes in the sense of T.G.Ostrom and to study the transofrmation method of designs introduced in [L.A.Rosati, P.Quattrocchi "Transformation of designs and other incidence structres" Geom. Ded. 44 (1992), 233-240].

1996 - Arcs in Minkowski planes [Articolo su rivista]
Rinaldi, Gloria; Quattrocchi, Pasquale
abstract

Some properties of k-arcs embedded in finite Minkoeski planes are studied, with a particular attention on problems of existence and completeness.

1996 - Arcs in the Hall Planes [Articolo su rivista]
Rinaldi, Gloria
abstract

Using a transformation tecnique for designs introduced in [P.Quattrocchi, L.A.Rosati "Transformation of designs and other incidence structures" Geom. Ded. (1992), 233-240], a class of arcs embeddable both in the Hall plane of order q^2 (q a prime power) and in its dual is constructedand. These arcs are complete in the unital of Gruning.

1995 - ALGEBRA [Monografia/Trattato scientifico]
Quattrocchi, Pasquale; Rinaldi, Gloria
abstract

Libro di testo per studenti del corso di laurea in Matematica.

1995 - Construction of Unitals in the Hall Planes [Articolo su rivista]
Rinaldi, Gloria
abstract

Using a transformation method for incidence structures introduced in [P.Quattrocchi, L.A.Rosati "Transformation of designs and other incidence structures" Geom. ded. 44 (1992), 233-240] I construct unitals embedded in the Hall planes by transformation of the Buekenhout-Metz unitals.

1995 - Hyperbolic unitals in the Hall planes [Articolo su rivista]
Rinaldi, Gloria
abstract

Using the transformation tecnique introduced in [P. Quattrocchi, L.A. Rosati "Transformation of designs and other incidence structres" Geom. Ded. (1992), 233-240], some sufficient conditions to transform a unital embedded in a projective plane into another one are given. As application unitals in the Hall planes are constructed by transformation of the hermitian curves. Necessary and sufficient conditions for the constructed unitals to be projectively equivalent are given too and fferent classes of not projectively equivalent Buekenhout's unitals are found in this manner. The unital of Gruning in the Hall plane is reconstructed and its embeddability in the dual of the Hall plane is also proved. Finally it is proved that the affine unital associated to the unital of Gruning is ismorphic to the hyperbolic hermitian curve.

1995 - Permutazioni e Geometrie [Articolo su rivista]
Fiori, Carla; Quattrocchi, Pasquale; Rinaldi, Gloria
abstract

1995 - Trasformazione di piani affini e insiemi di permutazioni strettamente 2-transitivi planari [Articolo su rivista]
Rinaldi, Gloria; E., Barozzi
abstract

In [L.A. Rosati and P.Quattrocchi "Transformation of designs and other incidence structures" geom ded., 44 (1992), pp.233-240] a procedure to transform incidence structures is introduced. Applying this procedure to finite affine planes one obtains an affine plane of the same order. While, when the procedure is applyied to a non finite affine plane a linear incidence structure is obtained but the axiom of parallelism is not assured. In this work we analyze sufficient conditions to transform not finite affine planes into affine planes and not finite planar sharply 2-transitive permutation sets into sharply 2-transitive permutation sets which are still planar.

1993 - Transformation of incidence structures and sharply multiply transitive permutation sets [Articolo su rivista]
Rinaldi, Gloria
abstract

The transformation process introduced in [P. Quattrocchi, L.A.Rosati "Transformation of designs and other incidence structures" Geom. ded. (1992), 233-240] is generalized. This generalization allows to construct examples of non-planar nearfileds and to construct the class of finite André planes by transformation of the desarguesian ones.

1993 - Transformation of multiply transitive permutation sets and finite regular near-fields [Articolo su rivista]
Rinaldi, Gloria
abstract

It is proved that a finite regular nearfield can be obtained by transformation of a field, in the sense of [L.A.Rosati, P.Quattrocchi "Transformations of designs and other incidence structures" Geom Ded. 44, (1992), 167-173] iff it is an André system.

1993 - Una caratterizzazione dei piani affini pappiani [Articolo su rivista]
Quattrocchi, Pasquale; Rinaldi, Gloria
abstract

Vengono presentati un insieme di assiomi che permettono di indentificare una struttura di incidenza come piano affine pappiano, ovvero coordinatizzabile tramite un campo. Gli assiomi e le dimostrazioni hanno carattere elementare.

1992 - Laguerre Semi-planes [Articolo su rivista]
Rinaldi, Gloria
abstract

Laguerre Semi-planes are defined and investigated.

1991 - Automorphisms of B-Geometries [Articolo su rivista]
Quattrocchi, Pasquale; Rinaldi, Gloria
abstract

(B)-Geometries are incidence structures arising from permutation sets. The automorphism groups of (B)-Geometries are studied in the paper. In certain cases they yield examples of inversive planes and sublplanes which are embedded in Minkowski planes. The automorphism groups of (B)-geometries associated with both the finite affine semilinear group and the finite projective semilineart group, in their representation on the points of the finite projective and affine line, are described.

1991 - Weakly Regular and Regular sets in Minkowski planes [Articolo su rivista]
Fiori, Carla; Rinaldi, Gloria
abstract

A regular (respectively weakly regular) set for an incidence structure Q is a set S of points such that the identity is the unique automorphism of Q which maps S onto iteslf (respectively which fixes S pointwise). Regular and weakly regular sets in finite Minkowski planes are investigated in the paper.

1990 - A characterization of PGL(2,q), q odd [Articolo su rivista]
Rinaldi, Gloria
abstract

A characterization of the projective linear group PGL(2,q) is given in term of involutions.

1990 - Standard spines of 3 - manifolds [Articolo su rivista]
Bandieri, Paola; Rinaldi, Gloria
abstract

In this paper, we present some relations between standard spines and crystallizations of closed, connected 3-manifolds.

1989 - Minkowski near-planes [Articolo su rivista]
Rinaldi, Gloria
abstract

Finite Minkowski near-planes are defined and investigated. It is proved that each finite Minkowski near-plane of order n with n at least 5, is either a Minkowski plane of order n-1 or it is embeddable in a Minkowski plane of order n. Minkowski near-planes of order 3 and 4 respectively are also descripted.

1989 - Minkowski semi-planes [Articolo su rivista]
Rinaldi, Gloria
abstract

Finite Minkowski semi-planes are definied and investigated. A complete classification is given.

1988 - Finite tangent circle structures [Relazione in Atti di Convegno]
P., Quattrocchi; Rinaldi, Gloria
abstract

Tangent-circle structures are incidence structures which comprehend projective and affine planes, Mobius planes and Laguerre planes. Different classes of finite tangent-circle structures are studied in the paper with a particular attention to those which are substructures of Laguerre and Mobius planes.