Partial graph design embeddings and related problems

Abstract

An embedding of a partial G-design (X, P) is a G-design (V, B) with the property that X C V and P C B. In 1976, Wilson [4] showed, among other things, that any partial G-design can be finitely embedded; however, the order of Wilson's embedding is exponentially large with respect to the order of the partial design. The problem of finding small embeddings of partial G-designs for various graphs G has received much attention in recent years, particularly for the case of partial Steiner triple systems and partial m-cycle systems with m ^ 4. The work of Andersen, Hilton, Mendelsohn, Hoffman, Lindner, Rodger, Bryant and others has seen considerable progress made on this problem. In contrast to this, the search for even a polynomial embedding of partial (n, k, l)-BIBDs, for any given k ^ 4, has been unsuccessful. The closest result of interest to this problem is Hoffman and Lindner's 8n + 1 6^ + 82 embedding for partial (K 4 \ i^2)-designs [2]. While the graph K t \ K 2 differs from K 4 (a block of size 4) by only one edge, Hoffman and Lindner's embedding relies heavily on the fact that AT 4 \ K 2 is tripartite. Since /f 4 is not tripartite, a small embedding for partial AT 4 -designs (that is, partial (n,4,1) BIBDs) appears to be beyond the reach of the current methods.In this thesis, we consider the problem of finding small embeddings of partial Gdesigns for various graphs G, as well as several closely related problems.Given the attention the embedding problem has received for the case of cycles, the initial focus of this thesis is on constructing small embeddings for certain cycle-related graphs G. In Chapter 2, we consider the case when G is either D § (a diagonal cycle of order 6) or a (4, l)-kite; note that both of these graphs are bipartite and differ from an even cycle by only one edge. Chapter 3 presents an embedding of partial (m, fc)-kite designs for any given values of m and k.In Chapter 4, a technique is presented for obtaining a cubic embedding of a restricted class of partial ^-designs. This embedding is valid for any partial AVdesign which has the property that every copy of K\ contains at least two vertices which do not occur in any other copy of K A .

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

“…Theorem 4.8. [31,32] An idempotent PLS(t), t 3, can be embedded in an idempotent LS((2t + 1) 2 ), which has an idempotent orthogonal mate.…”

“…Jenkins returned to PLS and used Evans' result (Theorem 3.3), to embed a PLS(t) in an LS(n), where n 2t, and subsequently applied Theorem 4.5 to prove: Theorem 4.7. [31,32] If t 4, then a PLS(t) can be embedded in an LS(4t 2 ) which has an orthogonal mate.…”