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Matchings in matroids over abelian groups

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Abstract

We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group \((G,+)\) is a bijection \(f:A\rightarrow B\) between two finite subsets A, B of G satisfying \(a+f(a)\notin A\) for all \(a\in A\). A group G has the matching property if for every two finite subsets \(A,B \subset G\) of the same size with \(0 \notin B\), there exists a matching from A to B. In Losonczy (Adv Appl Math 20(3):385–391, 1998) it was proved that an abelian group has the matching property if and only if it is torsion-free or cyclic of prime order. Here we consider a similar question in a matroid setting. We introduce an analogous notion of matching between matroids whose ground sets are subsets of an abelian group G, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.

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References

  1. Aharoni, R., Haxell, P.: Hall’s theorem for hypergraphs. J. Graph Theory 35(2), 83–88 (2000)

    Article  MathSciNet  Google Scholar 

  2. Aharoni, R., Kessler, O.: On a possible extension of Hall’s theorem to bipartite hypergraphs. Discrete Math. 84(3), 309–313 (1990)

    Article  MathSciNet  Google Scholar 

  3. Aliabadi, M., Filom, K.: Results and questions on matchings in abelian groups and vector subspaces of fields. J. Algebra 598, 85–104 (2022)

    Article  MathSciNet  Google Scholar 

  4. Aliabadi, M., Hadian, M., Jafari, A.: On matching property for groups and field extensions. J. Algebra Appl. 15(1), 1650011 (2016)

    Article  MathSciNet  Google Scholar 

  5. Aliabadi, M., Janardhanan, M.: On matchable subsets in abelian groups and their linear analogues. Linear Algebra Appl. 582, 138–155 (2019)

    Article  MathSciNet  Google Scholar 

  6. Aliabadi, M., Kinseth, J., Kunz, C., Serdarevic, H., Willis, C.: Conditions for matchability in groups and field extensions. Linear Multilinear A 71(7), 1182–1197 (2023)

    Article  MathSciNet  Google Scholar 

  7. Ardila-Mantilla, F. The geometry of geometries: matroid theory, old and new. ICM–International Congress of Mathematicians. Vol. VI, Sections 12–14, 4510–4541, EMS Press, Berlin, (2023)

  8. Alon, N., Fan, C.K., Kleitman, D., Losonczy, J.: Acyclic matchings. Adv. Math. 122(2), 234–236 (1996)

    Article  MathSciNet  Google Scholar 

  9. Bonin, J.E.: Basis-exchange properties of sparse paving matroids. Adv. in Appl. Math. 50, 6–15 (2013)

    Article  MathSciNet  Google Scholar 

  10. Crapo, H.H., Rota, G.C.: On the Foundations of Combinatorial Theory: Combinatorial Geometries. The M.I.T. Press, Cambridge (1970)

    Google Scholar 

  11. DeVos, M., Goddyn, L., Mohar, B.: A generalization of Kneser’s addition theorem. Adv. Math. 220(5), 1531–1548 (2009)

    Article  MathSciNet  Google Scholar 

  12. Eliahou, S., Lecouvey, C.: Matchings in arbitrary groups. Adv. in Appl. Math. 40, 219–224 (2008)

    Article  MathSciNet  Google Scholar 

  13. Eliahou, S., Lecouvey, C.: Matching subspaces in a field extension. J. Algebra 324, 3420–3430 (2010)

    Article  MathSciNet  Google Scholar 

  14. Fan, C.K., Losonczy, J.: Matchings and canonical forms for symmetric tensors. Adv. Math. 117(2), 228–238 (1996)

    Article  MathSciNet  Google Scholar 

  15. Ferroni, L., Nasr, G., Vecchi, L.: Stressed Hyperplanes and Kazhdan–Lusztig Gamma-positivity for matroids. Int. Math. Res. Not. rnac270 (2022)

  16. Hall, P.: On representatives of subsets. J. Lond. Math. Soc. 10, 26–30 (1935)

    Article  MathSciNet  Google Scholar 

  17. Hamidoune, Y.O.: Counting certain pairings in arbitrary groups. Combin. Probab. Comput. 20(6), 855–865 (2011)

    Article  MathSciNet  Google Scholar 

  18. Kemperman, J.H.B.: On complexes in a semigroup. Indag. Math. 18, 247–254 (1956)

    Article  MathSciNet  Google Scholar 

  19. Kemperman, J.H.B.: On small sumsets in an abelian group. Acta Math. 103, 63–88 (1960)

    Article  MathSciNet  Google Scholar 

  20. Lev, V.F.: The rectifiability threshold in abelian groups. Combinatorica 28(4), 491–497 (2008)

    Article  MathSciNet  Google Scholar 

  21. Levi, F.W.: Ordered groups. Proc. Indian Acad. Sci. 16, 256–263 (1942)

    Article  MathSciNet  Google Scholar 

  22. Losonczy, J.: On matchings in groups. Adv. in Appl. Math. 20(3), 385–391 (1998)

    Article  MathSciNet  Google Scholar 

  23. Mayhew, D., Newman, M., Welsh, D., Whittle, G.: On the asymptotic proportion of connected matroids. Eur. J. Combin. 32(6), 882–890 (2011)

    Article  MathSciNet  Google Scholar 

  24. Meshulam, R.: The clique complex and hypergraph matching. Combinatorica 21, 89–94 (2001)

    Article  MathSciNet  Google Scholar 

  25. Nakasawa, T.: Zur axiomatik der linearen abhängigkeit. I. Sci. Rep. Tokyo Bunrika Daigaku Sect. A 2(43), 235–255 (1935)

    Google Scholar 

  26. Nathanson, M.B.: Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Springer-Verlag, New York (1996)

    Book  Google Scholar 

  27. Oxley, J.: Matroid Theory, Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. Oxford University Press, Oxford (2011)

    Google Scholar 

  28. Pendavingh, R., van der Pol, J.: On the number of matroids compared to the number of sparse paving matroids. Electron. J. Combin. 22(2) (2015)

  29. Rado, R.: A theorem on independence relations. Quart. J. Math. Oxford Ser. 13, 83–89 (1942)

    Article  MathSciNet  Google Scholar 

  30. Schrijver, A., Seymour, P.D.: Spanning trees of different weights. DIMACS Ser. Discrete Math. Theoret. Comp. Sci. 1, 281–288 (1990)

    MathSciNet  Google Scholar 

  31. Wakeford, E.K.: On Canonical Forms. Proc. Lond. Math. Soc. (2) 18, 403–410 (1920)

    Article  MathSciNet  Google Scholar 

  32. Whitney, H.: On the abstract properties of linear dependence. Amer. J. Math. 57(3), 509–533 (1935)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are grateful to Shmuel Friedland for reading a preliminary version of this paper and for his useful suggestions. We also thank Richard Brualdi for drawing our attention to the various variants of the Hall’s marriage theorem. Finally, we are thankful to Steven J. Miller and David J Grynkiewicz for useful discussions on Kneser’s additive theorem.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, 9500 Gilman Dr, La Jolla, San Diego, CA, 92093, USA

    Mohsen Aliabadi

  2. Department of Mathematics, Iowa State University, 411 Morrill Road, Ames, IA, 50011, USA

    Shira Zerbib

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  1. Mohsen Aliabadi
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  2. Shira Zerbib
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Correspondence to Mohsen Aliabadi.

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Dedicated to Professor Shmuel Friedland on his 80th birthday.

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Shira Zerbib was supported by NSF Grant DMS-1953929

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Aliabadi, M., Zerbib, S. Matchings in matroids over abelian groups. J Algebr Comb (2024). https://doi.org/10.1007/s10801-024-01308-z

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