On the Algebraicity of the Tic-Tac-Toe Matroid and Homogeneous Mixed Bowtie Systems

This thesis is presented in two parts. First, we explore whether the class of algebraic matroids is closed under duality, a decades-old open question. We consider the Tic-Tac-Toe matroid as a potential candidate to answer the open question. The Tic-Tac-Toe matroid is known to satisfy many of the necessary conditions for a matroid to be algebraic and has a non-algebraic dual. Second, we focus on decompositions of the complete mixed graph into mixed bowties. A complete mixed graph has between every pair of vertices an undirected edge and antiparallel arcs. A mixed bowtie is a graph consisting of two 3-cycles which share exactly one vertex, and contains exactly four arcs and 2 edges. A decomposition of the complete mixed graph into mixed bowties is a mixed bowtie system. We provide necessary and sufficient conditions for the existence of all 12 homogeneous mixed bowtie systems and give constructions for each.