Beautifully Ordered Balanced Incomplete Block Designs and Embedding of Complements
Shealy, Andrew K., Jr.
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A brief description of known definitions and necessary conditions are presented along with some existence results. Chapter 1 will focus on Beautifully Ordered Balanced Incomplete Block Designs. Beautifully Ordered Balanced Incomplete Block Designs, BOBIBD(v,k,lambda,k 1, lambda1), were introduced by Chan and Sarvate along with some existence results for block size 3 and 4. We have shown that necessary conditions are sufficient for the existence of BOBIBDs with k = 5 for k1 = 2 and 3 along with partial results for k1 = 4. We also claim the nonexistence of cyclic solutions for certain BOBIBDs. The existence of the previously unknown BOBIBD(v,4,2,3,1), v ≡ 1(mod 6), is demonstrated for all v ≥ 19. Chapter 2 focuses on the concept of embedding designs. In particular, embedding of a balanced incomplete block design have been studied extensively but an embedding of a BIBD(v,k,lambda) into its complement BIBD(v, v-k, b-2r+lambda), v ≥ 2k, is not widely known. Embedding theorems as well as some examples of BIBDs where the embedding is not possible are presented, leading to the more general open question of finding the necessary and sufficient conditions for such embeddings.