Fulvio Zuanni

Bio: Fulvio Zuanni is an academic researcher from University of L'Aquila. The author has contributed to research in topics: Projective space & Order (group theory). The author has an hindex of 8, co-authored 28 publications receiving 183 citations.

{$G$}-invariantly resolvable Steiner {$2$}-designs which are {$1$}-rotational over {$G$}

Abstract: A 1-rotational (G;N;k; 1) dierence family is a set of k-subsets (base blocks) of an additive group G whose list of dierences covers exactly once G N and zero times N , N being a subgroup of G of order k 1. We say that such a dierence family is resolvable when the base blocks union is a system of representatives for the nontrivial right (or left) cosets of N in G. A Steiner 2-design is said to be 1-rotational over a group G if it admits G as an automorphism group xing one point and acting regularly on the remainder. We prove that such a Steiner 2-design is G-invariantly resolvable (i.e. it admits a G-invariant resolution) if and only if it is generated by a suitable 1-rotational resolvable dierence family over G.

Note on three-character (q+1)-sets in PG(3, q).

Abstract: We give a combinatorial characterization of twisted cubics in PG(3, q).

On two character ( q 7 + q 5 + q 2 + 1)-sets in PG(4, q 2 )

Abstract: This paper deals with (q7 + q5 + q2 + 1)-sets of type (m, n)3 in PG(4, q2), q > 2. Thus, with a minimal incidence condition with respect to dimension two, we obtain a combinatorial characterization of the non-singular Hermitian variety H (4, q2).

A survey on the existence of G-Designs

Abstract: A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G-designs. This includes references to all the necessary designs and constructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 373-410, 2008

Further progress on difference families with block size 4 or 5

Abstract: A strong indication about the existence of a (7p, 4, 1) difference family with p ? 7 (mod 12) a prime has been given in [11]. Here, developing some ideas of that paper, we give, much more generally, a strong indication about the existence of a cyclic (pq, 4, 1) difference family whenever p and q are primes congruent to 7 (mod 12) and of a cyclic (pq, 5, 1) difference family whenever p and q are primes congruent to 11 (mod 20). Indeed we give an algorithm for their construction that seems to be always successful and we have checked it works whenever both primes p and q do not exceed 1,000. All our (pq, 4, 1) and (pq, 5, 1) difference families have the nice property of admitting a multiplier of order 3 or 5, respectively, that fixes almost all base blocks. As an intermediate result we also find an optimal (p, 5, 1) optical orthogonal code for every prime p ? 11 (mod 20) not exceeding 10,000.

Strong difference families over arbitrary graphs

Abstract: The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406–425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply-vertex-transitive Γ-decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply-vertex-transitive Γ-decomposition of Km × e for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α-labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism T5 or the wheel W6. We also show that sometimes strong difference families lead to regular Γ-decompositions of a complete graph. We construct, for instance, a regular cube-decomposition of K16m for any integer m whose prime factors are all congruent to 1 modulo 6. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 443–461, 2008

1-Rotational k-Factorizations of the Complete Graph and New Solutions to the Oberwolfach Problem

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 of k = 2, a case in which we prove that 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 previously unknown solutions to the Oberwolfach problem via some direct and recursive constructions. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 87–100, 2008