TBSs in some minimum coverings

AbstractLet (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=0̸). The blocking set T is called tight if further V(Γi)∩T≠0̸ and V(Γi)∩(X∖T)≠0̸ 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

Anti-Pasch optimal coverings with triples

It is shown that for $v\ne 7,8,11,12$ or $13$, there exists an optimal covering with triples on $v$ points that contains no Pasch configurations

Multipartite graph decomposition: cycles and closed trails

This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not <em>K</em>_<em>n</em> ), in the case that the cycle lengths are “small”. Cycles up to length <em>n</em> are considered, when the complete multipartite graph has <em>n</em> parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned.<br /

Multipartite graph decomposition: cycles and closed trails

This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not K_n ), in the case that the cycle lengths are “small”. Cycles up to length n are considered, when the complete multipartite graph has n parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned

Generalized packing designs

Generalized $t$-designs, which form a common generalization of objects such as $t$-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of $t$-designs, \emph{Discrete Math.}\ {\bf 309} (2009), 4835--4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with $t=2$ and block size $k=3$ or 4.Comment: 38 pages, 2 figures, 5 tables, 2 appendices. Presented at 23rd British Combinatorial Conference, July 201

Cyclic packing designs and simple cyclic leaves constructed from Skolem-type sequences

A Packing Design, or a PD(v; k; λ) is a pair (V; β) where V is a v-set of points and β is a set of k-subsets (blocks) such that any 2-subset of V appears in at most λ blocks. PD(v; k; λ) is cyclic if its automorphism group contains a v-cycle, and it is called a cyclic packing design. The edges in the multigraph λKᵥ not contained in the packing form the leaves of the CPD(v; k; λ); denoted by leave (v; k; λ) : In 2012, Silvesan and Shalaby used Skolem-type sequences to provide a complete proof for the existence of cyclic BIBD(v; 3; λ) for all admissible orders v and λ. In this thesis, we use Skolem-type sequences to find all cyclic packing designs with block size 3 for a cyclic BIBD(v; 3; λ) and find the spectrum of leaves graph of the cyclic packing designs, for all admissible orders v and λ with the optimal leaves, as well as determine the number of base blocks for every λ when k = 3