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The Graphs And Matroids Whose Only Odd Circuits Are Small, Kristen Nicole Wetzler
The Graphs And Matroids Whose Only Odd Circuits Are Small, Kristen Nicole Wetzler
LSU Doctoral Dissertations
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 …