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The Merino-Welsh conjecture for split matroids. (English) Zbl 1526.05027

Summary: C. Merino and D. J. A. Welsh [ibid. 3, No. 2–4, 417–429 (1999; Zbl 0936.05043)] conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices
05C31 Graph polynomials
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)

Citations:

Zbl 0936.05043

Software:

SageMath

References:

[1] K. Bérczi, T. Király, T. Schwarcz, Y. Yamaguchi, and Y. Yokoi. Hypergraph characterization of split matroids. J. Combin. Theory Ser. A, 194:Paper No. 105697, 2023. · Zbl 1502.05026
[2] Bonin, JE; de Mier, A., The lattice of cyclic flats of a matroid, Ann. Comb., 12, 2, 155-170 (2008) · Zbl 1145.05015 · doi:10.1007/s00026-008-0344-3
[3] T. Brylawski and J. Oxley. The Tutte polynomial and its applications. In Matroid applications, volume 40 of Encyclopedia Math. Appl., pages 123-225. Cambridge Univ. Press, Cambridge, 1992. · Zbl 0769.05026
[4] Cameron, A.; Mayhew, D., Excluded minors for the class of split matroids, Australas. J. Combin., 79, 195-204 (2021) · Zbl 1465.05173
[5] Chávez-Lomelí, LE; Merino, C.; Noble, SD; Ramírez-Ibáñez, M., Some inequalities for the Tutte polynomial, European J. Combin., 32, 3, 422-433 (2011) · Zbl 1290.05055 · doi:10.1016/j.ejc.2010.11.005
[6] Conde, R.; Merino, C., Comparing the number of acyclic and totally cyclic orientations with that of spanning trees of a graph, Int. J. Math. Comb., 2, 79-89 (2009) · Zbl 1198.05015
[7] G. W. Dinolt. An extremal problem for non-separable matroids. In Théorie des matroïdes (Rencontre Franco-Britannique, Brest, 1970), pages 31-49. Lecture Notes in Math. Vol. 211. 1971. · Zbl 0215.33603
[8] Ferroni, L., On the Ehrhart polynomial of minimal matroids, Discrete Comput. Geom., 68, 1, 255-273 (2022) · Zbl 1490.05026 · doi:10.1007/s00454-021-00313-4
[9] L. Ferroni and B. Schröter. Valuative invariants for large classes of matroids. arXiv e-prints, page arXiv:2208.04893, Aug. 2022.
[10] Jackson, B., An inequality for Tutte polynomials, Combinatorica, 30, 1, 69-81 (2010) · Zbl 1225.05135 · doi:10.1007/s00493-010-2484-4
[11] Joswig, M.; Schröter, B., Matroids from hypersimplex splits, J. Combin. Theory Ser. A, 151, 254-284 (2017) · Zbl 1366.05024 · doi:10.1016/j.jcta.2017.05.001
[12] Knauer, K.; Martínez-Sandoval, L.; Ramírez Alfonsín, JL, A Tutte polynomial inequality for lattice path matroids, Adv. in Appl. Math., 94, 23-38 (2018) · Zbl 1377.05089 · doi:10.1016/j.aam.2016.11.008
[13] Mayhew, D.; Newman, M.; Welsh, D.; Whittle, G., On the asymptotic proportion of connected matroids, European J. Combin., 32, 6, 882-890 (2011) · Zbl 1244.05047 · doi:10.1016/j.ejc.2011.01.016
[14] Merino, C.; Welsh, DJA, Forests, colorings and acyclic orientations of the square lattice, Ann. Comb., 3, 2-4, 417-429 (1999) · Zbl 0936.05043 · doi:10.1007/BF01608795
[15] Noble, SD; Royle, GF, The Merino-Welsh conjecture holds for series-parallel graphs, European J. Combin., 38, 24-35 (2014) · Zbl 1282.05031 · doi:10.1016/j.ejc.2013.11.002
[16] J. Oxley. Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, second edition, 2011. · Zbl 1254.05002
[17] The Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.1), 2020.
[18] Thomassen, C., Spanning trees and orientations of graphs, J. Comb., 1, 2, 101-111 (2010) · Zbl 1219.05044
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