Circular Flow and Circular Chromatic Number in the Matroid Context

This thesis considers circular flow-type and circular chromatic-type parameters (φ and χ, respectively) for matroids. In particular we focus on orientable matroids and 6 √ 1-matroids. These parameters are obtained via two approaches: algebraic and orientation-based. The general questions we discuss are: bounds for flow number; characterizations of Eulerian and bipartite matroids; and possible connections between the two possible extensions of φ: algebraic and orientation. In the case of orientable matroids, we obtain characterizations of bipartite rank-3 matroids and Eulerian uniform, rank-3 matroids; an asymptotic result regarding the flow number of uniform matroids; and an improvement on the known bound for flow number of matroids of arbitrary rank. This bound is further improved for the uniform case. For 6 √ 1-matroids, we examine an algebraic extension of the parameters χ and φ. We also introduce a notion of orientation and the corresponding flow and chromatic numbers applicable to this class. We investigate the possibility of a connection between the algebraic and orientation-based parameters, akin to that established for regular matroids by Hoffman’s Circulation Lemma and we obtain a partial connection. We extend the notion of “Eulerian” to 6 √ 1-matroids. We call such matroids