Decompositions of complete graphs into bipartite 2-regular subgraphs

It is shown that if G is any bipartite 2-regular graph of order at most n/2 or at least n – 2, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order n into a perfect matching and edge-disjoint copies of G

Decomposing the blocks of a Steiner triple system of order 4v-3 into partial parallel classes of size v-1

In this report we present a summary and our new results on finding partial parallel classes of uniform size of Steiner triple systems, STS(v). We show several results for STS(4v - 3), where v = 3 mod 12 and v = 9 mod 12. In Chapter 1 we provide background knowledge and introduce the problem. In Chapter 2 we discuss some important known results to the problem, introduce the needed ingredients, and explain the methodology of the construction. Finally, in Chapter 3, we conclude with a summary and discuss possibilities for future work

Maximum Kirkman Signal Sets for Synchronous Uni-Polar Multi-User Communication Systems

A unipolar signaling system transmits using intensity or amplitude in multiple dimensions. Typical examples arise in optical transmission or radio communication using MT-MFSK as both the signaling and the modulation technique. There are v dimensions which represent pulses or tones. Each codeword consists of a selection of k of these tones with unit intensity. Each user is assigned m of these binary codewords. In a synchronous multi-user environment, two codewords assigned to a single user have distance 2k, while two codewords assigned to different users have distance at least 2k \Gamma 2. Such an assignment of codewords to users is called a Kirkman signal set when the number of users accommodated is the maximum. In this paper, the existence of Kirkman signal sets with k = 3 and m as large as possible is settled for all values of v. 1 Introduction A balanced incomplete block design (BIBD) is a pair (V; B) where V is a v-set and B is a collection of b k-subsets of V (blocks) such that..

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