Designs for graphs with six vertices and ten edges - II

The design spectrum has been determined for ten of the 15 graphs with six vertices and ten edges. In this paper, we solve the design spectrum problem for the remaining five graphs with three possible exceptions

On the study of balanced incomplete block designs with repeated blocks and other incomplete block designs

Not availabl

The 3-GDDs of type g3u2g^3u^2

A 3-GDD of type ${g^3u^2}$ exists if and only if $g$ and $u$ have the same parity, $3$ divides $u$ and $u\leq 3g$.Such a 3-GDD of type ${g^3u^2}$ is equivalent to an edge decomposition of $K_{g,g,g,u,u}$ into triangles

Sets of three pairwise orthogonal Steiner triple systems

AbstractTwo Steiner triple systems (STS) are orthogonal if their sets of triples are disjoint, and two disjoint pairs of points defining intersecting triples in one system fail to do so in the other. In 1994, it was shown (Canad. J. Math. 46(2) (1994) 239–252) that there exist a pair of orthogonal Steiner triple systems of order v for all v≡1,3 (mod6), with v⩾7, v≠9. In this paper we show that there exist three pairwise orthogonal Steiner triple systems of order v for all v≡1(mod6), with v⩾19 and for all v≡3(mod6), with v⩾27 with only 24 possible exceptions

The Hadamard Matroid and an Anomaly in Its Single Element Extensions

A nonstandard vector space is formulated, whose bases afford a representation of what is called a Hadamard matroid, Mp. For prime p, existence of Mp is equivalent to the existence of both a classical Hadamard matrix H(p,p) and a certain affine resolvable, balanced incomplete block design AR(p). An anomaly in the representable single element extension of a Hadamard matroid is discussed

Edge coloring BIBDS and constructing MOELRs

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Some historical uses and background are touched upon as well. The majority of the definitions are contained within this chapter as well. In Chapter 2 we consider the question whether one can decompose λ copies of monochromatic Kv into copies of Kk such that each copy of the Kk contains at most one edge from each Kv. This is called a proper edge coloring (Hurd, Sarvate, [29]). The majority of the content in this section is a wide variety of examples to explain the constructions used in Chapters 3 and 4. In Chapters 3 and 4 we investigate how to properly color BIBD(v, k, λ) for k = 4, and 5. Not only will there be direct constructions of relatively small BIBDs, we also prove some generalized constructions used within. In Chapter 5 we talk about an alternate solution to Chapters 3 and 4. A purely graph theoretical solution using matchings, augmenting paths, and theorems about the edgechromatic number is used to develop a theorem that than covers all possible cases. We also discuss how this method performed compared to the methods in Chapters 3 and 4. In Chapter 6, we switch topics to Latin rectangles that have the same number of symbols and an equivalent sized matrix to Latin squares. Suppose ab = n2. We define an equitable Latin rectangle as an a × b matrix on a set of n symbols where each symbol appears either [b/n] or [b/n] times in each row of the matrix and either [a/n] or [a/n] times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of ka × b mutually orthogonal equitable Latin rectangles as a k–MOELR(a, b; n). We show that there exists a k–MOELR(a, b; n) for all a, b, n where k is at least 3 with some exceptions