Alberto Del Fra

Cyclic Hamiltonian cycle systems of the complete graph

Abstract We prove that there exists a cyclic Hamiltonian k-cycle system of the complete graph if and only if k is odd but k≠15 and pα with p prime and α>1. As a consequence we have the existence of a cyclic k-cycle system of the complete graph on km vertices for any pair (k,m) of odd integers with k as above but (k,m)≠(3,3).

Existence of cyclic k -cycle systems of the complete graph

Starting from earliest papers by Rosa we solve, directly and explicitly, the existence problem for cyclic k-cycle systems of the complete graph Kv with v ≡ 1 (mod 2k), and the existence problem for cyclic k-cycle systems of the complete m-partite graph Km×k with m and k being odd. As a particular consequence, a cyclic p-cycle system of Kv with p being a prime exists for all admissible values of v but (p,v) ≠ (3,9). This was previously known only for p = 3, 5, 7.

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.Abstractd-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.

Dimensional dual hyperovals associated with Steiner systems

In Adv. Geom. 3 (2003) 245, a class of d-dimensional dual hyperovals is constructed starting from a subset X of PG(d, 2) with certain properties. In this paper, a criterion for X to provide a d-dimensional dual hyperoval is given in terms of some functions. Based on this, we describe such subsets, and show that there are exactly two isomorphism classes of d-dimensional dual hyperovals arising from those subsets and that a similar statement holds for those of the associated Steiner systems S(3, 4, 2d+1).

On the span of polynomials with integer coefficients

Following a paper of R. Robinson, we classify all hyperbolic polynomials in one variable with integer coefficients and span less than 4 up to degree 14, and with some additional hypotheses, up to degree 17. We conjecture that the classification is also complete for degrees 15, 16, and 17. Besides improving on the method used by Robinson, we develop new techniques that turn out to be of some interest. A close inspection of the polynomials thus obtained shows some properties deserving further investigations.

Extended partial geometries with minimal μ

Recently Fisher and Hobart extended the original work of Buekenhout and Hubaut on extended generalized quadrangles with a certain minimal value (‘minimal μ’) for the nonempty residue intersections. We show that there is another minimal μ which is sometimes better than the earlier one, and we extend all this to the case of extended partial geometries. We examine especially the triangular case, and find only a few new possibilities, which we are unable to settle, although some of these look extremely interesting.

Classification of extended generalized quadrangles with maximum diameter

Let S be an extended generalized quadrungle of order (s, t). Recently it has been proved that the diameter Δ of the point-graph S satisfies Δ⩽s + 1 (see Cameron, Hughes and Pasini [4]). In this paper we prove that Δ = s + 1 if and only if one of the following occurs: 1. (i) t = 1 and S is isomorphic to the Johnson geometry on (2(s+1)s+1) points (s > 0). 2. (ii) s = 2, t = 2, 4 and S is isomorphic to the affine polar space of order 2 and type A2, D-2 on 32 and 56 points respectively. 3. (iii) s = 1 and S is completely tripartite on 3(t + 1) points (t > 1).

On two new classes of semibiplanes

Abstract Two new infinite classes of semibiplanes are defined by considering for any q = 2 h , h > 1, a point-plane pair ( p , π ) in PG(3, q ), a hyperoval O in the star of p and a dual hyperoval O ∗ in the plane π in suitable mutual position. These geometries are called of flag type or anti-flag type according to p and π are incident or not. By deleting some suitable elements from the semibiplanes of flag type we obtain another family of semibiplanes. In all of cases some quotients are defined. For q = 4, the semibiplane of anti-flag type is a flag-transitive geometry already given by Pasini and Yoshiara.

TruncatedDn coxeter complexes as extended partial geometries with maximum diameter

In [9] we proved that the diameter Δ of an Extended partial Geometry (EpGα), of order (s,t) with α ≥2 is bounded by [s/2] −ϕ + 4, where ϕ is the index of the geometry (i.e. the minimum positive antiflag number). An example of EpG2(s,1) with ϕ=α + 1 = 3 attaining the bound is given by a truncated Ds+2 Coxeter complex (see Example 1.2 for a different simple description).In this paper we prove that ifs is even, anEpGα with diameter Δ=[s/2] −ϕ+ 4 > 3, sayS, has necessarily α = 2,s ≥ 6 andt = 1; if furthermore ϕ=α + 1, thenS is isomorphic to Ds+2.In the cases odd, we give a characterization which still suggests the conjecture that Ds+2 is the unique example (apart from few sporadic cases).The classification of Extended Generalized Quadrangles (EGQ) with maximum diameter was given in [8]. In that case (which correspond to α=1) the unique example (apart from few sporadic cases) was given by the Johnson geometry which can also be described by a truncation of a Coxeter complex of type A2s+1