Counterexamples to conjectures on 4-connected matroids

Abstract

Tutte characterized binary matroids to be those matroids without aU 24 minor. Bixby strengthened Tutte’s result, proving that each element of a 2-connected non-binary matroid is in someU 24 minor. Seymour proved that each pair of elements in a 3-connected non-binary matroid is in someU 24 minor and conjectured that each triple of elements in a 4-connected non-binary matroid is in someU 24 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.

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Coullard, C.R. Counterexamples to conjectures on 4-connected matroids. Combinatorica 6, 315–320 (1986). https://doi.org/10.1007/BF02579257

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AMS subject classification (1980)

  • 05 B 35