Bounding the beta invariant of 3-connected matroids

Abstract The beta invariant is related to the Chromatic and Tutte Polynomials and has been studied by Crapo [4] , Brylawski [2] , Oxley [7] and others. Crapo [4] showed that a matroid with at least two elements is connected if and only if its beta invariant is greater than zero. Brylawski [2] showed that a connected matroid has beta invariant one if and only if M is isomorphic to a serial-parallel network. Oxley [7] characterized all matroids with beta invariant two, three and four. In this paper, we first give a best possible lower bound on the beta invariant of 3-connected matroids, then we characterize all 3-connected matroids attaining the lower bound. We also characterize all binary matroids with beta invariant 5, 6, and 7.