Abstract
A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzík proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r ≥ 3, no rank−r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.
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References
Bachem A., Kern W.: On sticky matroids. Discrete Math. 69(1), 11–18 (1988)
Bonin J., de Mier A.: The lattice of cyclic flats of a matroid. Ann. Combin. 12(2), 155–170 (2008)
Brylawski T.H.: An affine representation for transversal geometries. Stud. Appl. Math. 54(2), 143–160 (1975)
Crapo H.H.: Single-element extensions of matroids. J. Res. Nat. Bur. Standards Sect. B 69B, 55–65 (1965)
Kantor W.: Dimension and embedding theorems for geometric lattices. J. Combin. Theory Ser. A 17, 173–195 (1974)
Oxley J.G.: Matroid Theory. Oxford University Press, Oxford (1992)
Poljak S., Turzík D.: A note on sticky matroids. Discrete Math. 42(1), 119–123 (1982)
Sims, J.A.: Some Problems in Matroid Theory. Ph.D. Dissertation, Linacre College, Oxford University, Oxford (1980)
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Bonin, J.E. A Note on the Sticky Matroid Conjecture. Ann. Comb. 15, 619–624 (2011). https://doi.org/10.1007/s00026-011-0112-7
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DOI: https://doi.org/10.1007/s00026-011-0112-7