Counterexamples to conjectures on 4-connected matroids

Abstract

Tutte characterized binary matroids to be those matroids without aU 4 2 minor. Bixby strengthened Tutte’s result, proving that each element of a 2-connected non-binary matroid is in someU 4 2 minor. Seymour proved that each pair of elements in a 3-connected non-binary matroid is in someU 4 2 minor and conjectured that each triple of elements in a 4-connected non-binary matroid is in someU 4 2 minor. A related conjecture of Robertson is that each triple of elements in a 4-connected non-graphic matroid is in some circuit. This paper provides counterexamples to these two conjectures.