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 propertynthat X C V and P C B. In 1976, R.M. Wilson [66] showed, among other things,nthat any partial G-design can be finitely embedded; however, the order of Wilson'snembedding is exponentially large with respect to the order of the partial design.nThe problem of finding small embeddings of partial G-designs for various graphsnG has received much attention in recent years, particularly for the case of partialnSteiner triple systems and partial m-cycle systems with m g 4. The work of Andersen,nHilton, Mendelsohn, Hoffman, Lindner, Rodger, Bryant and others has seennconsiderable progress made on this problem. In contrast to this, the search for evenna polynomial embedding of partial (n, k, 1)-BIBDs, for any given k g 4, has beennunsuccessful. The closest result of interest to this problem is Hoffman and Lindner'sn8n + 16rn + 82 embedding for partial (K4 K2)-designs [26]. While the graphnK4 K2 differs from K4 (a block of size 4) by only one edge, Hoffman and Lindner'snembedding relies heavily on the fact that K4 K2 is tripartite. Since K4 is notntripartite, a small embedding for partial K4-designs (that is, partial (n, 4, 1) BIBDs)nappears to be beyond the reach of the current methods.n n In this thesis, we consider the problem of finding small embeddings of partialnG-designs for various graphs G, as well as several closely related problems.n n Given the attention the embedding problem has received for the case of cycles,nthe initial focus of this thesis is on constructing small embeddings for certain cycle-relatedngraphs G. In Chapter 2, we consider the case when G is either D6 (a diagonalncycle of order 6) or a (4, 1)-kite; note that both of these graphs are bipartite andndiffer from an even cycle by only one edge. Chapter 3 presents an embedding ofnpartial (m, k)-kite designs for any given values of m and k.n n In Chapter 4, a technique is presented for obtaining a cubic embedding of anrestricted class of partial K4-designs. This embedding is valid for any partial K4-ndesign which has the property that every copy of K4 contains at least two verticesnwhich do not occur in any other copy of K4. n The concept of irreducibility is introduced in Chapter 5, and is applied to 2-nfold m-cycle systems. Specifically, it is shown that for any pair of integers (m, n)nwith 4 l m l n, if there exists an m-cycle system of order n, then there exists annirreducible 2-fold m-cycle system of order n, except when (m, n) = (5, 5). A similarnresult has already been established by Kramer [35] for the case of 3 -cycles.n n Chapter 6 describes further results on irreducible l-fold m-cycle systems when lg 2. We prove the existence of irreducible 3 -fold m-cycle systems of various orders,nand show that for any l g 2, there exists an irreducible A-fold m-cycle system ofnorder n for a sufficiently large integer n.n n The problem of immersing partial Steiner triple systems is the focus of Chapter 7.nAn immersing of a partial Steiner triple system (V, P) of order n is a l-fold triplensystem (V, B) of order n such that P C B. It is shown that a partial Steiner triplensystem of order n can always be immersed in a 6-fold triple system of order n if nnis odd, and a 12 -fold triple system of order n if n is even.n n Finally, in Chapter 8 we turn our attention to pairs of orthogonal latin squares.nHeinrich and Zhu [24] have shown that a pair of orthogonal latin squares of ordernn can be embedded in a pair of orthogonal latin squares of order t if t g 3n,nthe bound of 3n being best possible. Obtaining an analogous result for pairs ofnpartial orthogonal latin squares has proved to be an extremely challenging problem.nAlthough Lindner has proved that a pair of partial orthogonal latin squares cannalways be finitely embedded, there is no known method which obtains an embeddingnof polynomial order (with respect to the order of the partial arrays). The lessndifficult problem of embedding a single partial latin square in a pair of orthogonalnlatin squares is investigated in this chapter.