Antoinette Tripodi

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.

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 } .

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 .

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.

Minimum embedding of a KS(u, λ) into a KS(u+w, µ)

Let H be a subgraph of a graph G. An H-design (U,C) of order u and index @l is embedded into a G-design (V,B) of order v and index @m if @l@?@m, U@?V and there is an injective mapping f:C->B such that B is a subgraph of f(B) for every B@?C. The mapping f is called the embedding of (U,C) into (V,B). In this paper, we study the minimum embedding of a kite system of order u and index @l (denoted by KS(u,@l)) into a kite system of order u+w and index @m.

Metamorphosis of simple twofold triple systems into twofold (K4−e)-designs

Abstract In this note, we show that there exists a metamorphosis of a simple twofold triple system of order v into a twofold ( K 4 − e ) -design of order v if and only if v ≡ 0 , 1 , 6 , 10 ( mod 15 ) and v ⩾ 10 .

TBSs in some minimum coverings

Let (X,B) be a (λKn,G)-covering with excess E and a blocking set T. Let Γ1, Γ2, …, Γs be all connected components of E with at least two vertices (note that s=0 if E=0). The blocking set T is called tight if further V(Γi)∩T≠0 and V(Γi)∩(X∖T)≠0 for 1≤i≤s. In this paper, we give a complete solution for the existence of a minimum (λKn,G)-covering admitting a blocking set (BS), or a tight blocking set (TBS) for any λ and when G=K3 and G=K3+e.

The Doyen-Wilson theorem for 3-sun systems

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.

The spectrum of α-resolvable λ-fold (K_4 - e)-designs

A λ -fold G -design is said to be α -resolvable if its blocks can be partitioned into classes such that every class contains each vertex exactly α times. In this paper we study the α -resolvability for λ -fold ( K 4  −  e ) -designs and prove that the necessary conditions for their existence are also sufficient, without any exception.

Minimum embedding of path designs into kite systems

Abstract Let G a simple graph and H be a subgraph of G, and let U ⊆ V . We say that a λ-fold H-design ( U , C ) of order u is embedded into a μ-fold G-design ( V , B ) of order u + w , if there is a injective function f : C → B such that B is a subgraph of f ( B ) for every B ∈ C . If f : C → B is bijective, the embedding is called exact. In this paper we solve the embedding problem and the exact embedding problem of a P k ( u , λ ) into a K S ( u , μ ) , with k = 2 , 3 , 4 .