Doctor of Philosophy (PhD)
This thesis is motivated by a graph-theoretical result of Maffray, which states that a 2-connected graph with no odd cycles exceeding length 3 is bipartite, is isomorphic to K_4, or is a collection of triangles glued together along a common edge. We first prove that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M is M(K_4) or F_7, or M is the cycle matroid of a graph consisting of a collection of triangles glued together along a common edge. This result implies that a 2-connected loopless graph G has no odd bonds of size at least five if and only if G is Eulerian or G is a subdivision of either K_4 or the graph that is obtained from a cycle of parallel pairs by deleting a single edge. The main theorem of the dissertation extends Maffray's theorem to n-connected graphs with no odd cycles exceeding size 2n-1. To prove this, we first prove the special cases when n=3 and n=4. The proof of the theorem is competed with an argument that treats all n>= 5.
Wetzler, Kristen Nicole, "The Graphs and Matroids Whose Only Odd Circuits Are Small" (2018). LSU Doctoral Dissertations. 4185.