首页|The binary matroids with no odd circuits of size exceeding five
The binary matroids with no odd circuits of size exceeding five
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NSTL
Elsevier
Generalizing a graph-theoretical result of Maffray to binary matroids, Oxley and Wetzler proved that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M is isomorphic to M(K-4) or F-7, or M is the cycle matroid of a graph consisting of a collection of triangles all sharing a common edge. In this paper, we show that if M is a 3-connected binary matroid having a 5-element circuit but no larger odd circuit, then M has rank less than six; or M has rank six and is one of nine sporadic matroids; or M can be obtained by attaching together, via generalized parallel connection across a common triangle, a collection of copies of F-7 and M(K-4) and then possibly deleting up to two elements of the common triangle. From this, we deduce that a 3-connected simple graph with a 5-cycle but no larger odd cycle is obtained from K-3,K-n for some n >= 3 by adding one, two, or three edges between the vertices in the 3-vertex class. (C) 2021 Elsevier Inc. All rights reserved.
NSTL
Elsevier
