Discrete Mathematics
Triads and triangles in 3-connected matroids
Discrete Mathematics
Graph Theory With Applications
Graph Theory With Applications
On Minimally 3-Connected Binary Matroids
Combinatorics, Probability and Computing
On the number of triangles in 3-connected matroids
European Journal of Combinatorics
Non-separating cocircuits in matroids
Discrete Applied Mathematics
Minimally 3-connected binary matroids
European Journal of Combinatorics
Obstructions to a binary matroid being graphic
European Journal of Combinatorics
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A cocircuit C* in a matroid M is said to be non-separating if and only if M∖C*, the deletion of C* from M, is connected. A vertex-triad in a matroid is a three-element non-separating cocircuit. Non-separating cocircuits in binary matroids correspond to vertices in graphs. Let C be a circuit of a 3-connected binary matroid M such that ∣E(M)∣≥4 and, for all elements x of C, the deletion of x from M is not 3-connected. We prove that C meets at least two vertex-triads of M. This gives direct binary matroid generalizations of certain graph results of Halin, Lemos, and Mader. For binary matroids, it also generalizes a result of Oxley. We also prove that a minimally 3-connected binary matroid M which has at least four elements has at least ½r*(M)+1 vertex-triads, where r*(M) is the corank of the matroid M. An immediate consequence of this result is the following result of Halin: a minimally 3-connected graph with n vertices has at least 2n+6/5 vertices of degree three. We also generalize Tutte's Triangle Lemma for general matroids.