Mathematical Sciences - Colloquium

Connectivity theory is one of the most important subjects in graph theory and matroid theory. There has been much interest in generalizing graph results to matroids, especially to binary matroids. In this talk, we will present several binary matroid connectivity results which generalize certain graph results. A very useful graph result of Mader states that if C is a cycle of a 3-connected graph G such that for all elements x of C, the deletion of x from G is not 3-connected, then C meets at least two vertices of degree three. We prove that this result is a special case of a connectivity result for binary matroids. Another well-known graph result of Halin states that a minimally 3-connected graph with n vertices has at least 2n+6 / 5 vertices of degree three. We prove a binary matroid generalization of this result. We use the concept of non-separating cocircuits in matroids.

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