Salvatore Milici

Colouring Steiner systems with specified block colour patterns

Abstract We consider colourings of Steiner systems S (2,3, v ) and S (2,4, v ) in which blocks have prescribed colour patterns, as a refinement of the classical weak colourings. The main question studied is, given an integer k, does there exist a colouring of given type using exactly k colours? For several types of colourings, a complete answer to this question is obtained while for other types, partial results are presented. We also discuss the question of the existence of uncolourable systems.

Embedding handcuffed designs with block size 2 or 3 in 4-cycle systems

Abstract Let (W, C ) be an m -cycle system of order n and let Ω ⊂W , | Ω |=v . We say that a handcuffed design ( Ω , P ) of order v and block size s ( 2⩽s⩽m−1 ) is contained in (W, C ) if for every p∈ P there is an m -cycle c=(a 1 ,a 2 ,…,a m )∈ C such that: (1) p=[a k ,a k+1 ,…,a k+s−1 ] for some k∈{1,2,…,m} (i.e. the (s−1) -path p occurs in the m -cycle c ); and (2) a k−1 ,a k+s ∉ Ω . Note that in (1) and (2) all the indices are reduced to the range {1,…,m} ( mod m) . For each n≡1 ( mod 8) and for each s∈{2,3} we determine all the integers v such that there is a 4 -cycle system of order n containing a handcuffed design of order v and block size s .

On the existence of uniformly resolvable decompositions of Kv and Kv−I into paths and kites

Abstract In this paper, it is shown that, for every v ≡ 0 ( mod 12 ) , there exists a uniformly resolvable decomposition of K v - I , the complete undirected graph minus a 1 -factor, into r classes containing only copies of 2-stars and s classes containing only copies of kites if and only if ( r , s ) ∈ { ( 3 x , 1 + v − 4 2 − 2 x ) , x = 0 , … , v − 4 4 } . It is also shown that a uniformly resolvable decomposition of K v into r classes containing only copies of 2-stars and s classes containing only copies of kites exists if and only if v ≡ 9 ( mod 12 ) and s = 0 .

Maximum uniformly resolvable decompositions of K v and K v - I into 3-stars and 3-cycles

Let K v denote the complete graph of order v and K v - I denote K v minus a 1-factor. In this article we investigate uniformly resolvable decompositions of K v and K v - I into r classes containing only copies of 3-stars and s classes containing only copies of 3-cycles. We completely determine the spectrum in the case where the number of resolution classes of 3-stars is maximum.

Octagon kite systems

Abstract The spectrum of octagon kite system (OKS) which is nesting strongly balanced 4-kite-designs is determined.

Disjoint blocking sets in cycle systems

Abstract In an m -cycle system C of order n (n⩾m⩾3 integers) , the blocks are the vertex sets of n(n−1)/(2m) cycles C i of length m such that each edge of the complete graph K n belongs to precisely one cycle C i ∈ C . We investigate m -cycle systems which admit vertex partitions into two or more classes in such a way that each class meets every cycle of C . Relatively small systems (with n⩽2 m /( e m) ) are always ‘2-colorable’ in this sense; moreover, for every constant c, if n⩽cm , then a partition into c′m/ log m classes exists (where the constant c′ depends only on c ).

Uniformly resolvable decompositions of Kv into P3 and K3 graphs

Abstract In this paper we consider the uniformly resolvable decompositions of the complete graph K v , or the complete graph minus a 1-factor as appropriate, 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 3 s or K 3 s.

Cycle systems without 2-colorings

In an m-cycle system C of order n the blocks are the vertex sets of n(n−1)/(2m) cycles Ci such that each edge of the complete graph Kn belongs to precisely one cycle Ci E C. We prove the existence of m-cycle systems that admit no vertex partition into two classes in such a way that each class meets every cycle of C. The proofs apply both constructive and probabilistic methods, and also some old and new facts about Steiner Triple Systems without large independent sets.

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 } .

Uniformly resolvable decompositions of K v into P 3 and K 3 graphs

Abstract In this paper we consider the uniformly resolvable decompositions of the complete graph K v , or the complete graph minus a 1-factor as appropriate, 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 3 s or K 3 s.