UVM Theses and Dissertations
Format:
Print
Author:
Brown, Melanie
Dept./Program:
Mathematics
Year:
2010
Degree:
PhD
Abstract:
The complete graph Kn triangulates some orientable surface if and only if n = 0,3,4, 7 (mod 12). These triangulations are particularly interesting when the embedding is face 2-colorable. In such an embedding, we can consider the three vertices of each triangular face as a triple, and these triples form a Steiner triple system on each of the two color classes. We call such an embedding a biembedding. The existence of biembeddings is known for all possible values of n.
We seek to expand these results by identifying families of combinatorial designs that pairwise biembed. We present a construction giving families of biembeddable Steiner triple systems for certain classes of n. We also consider biembeddings for families of group divisible designs of order 3n and type n³. Furthermore, we develop a recursive construction that embeds a Steiner triple system and a 4-cycle system on n points for n = gk for k>_1.
We seek to expand these results by identifying families of combinatorial designs that pairwise biembed. We present a construction giving families of biembeddable Steiner triple systems for certain classes of n. We also consider biembeddings for families of group divisible designs of order 3n and type n³. Furthermore, we develop a recursive construction that embeds a Steiner triple system and a 4-cycle system on n points for n = gk for k>_1.