Mathematica Pannonica
Volume/Issue:
Volume 29_NS3: Issue 1
Perfect Solutions to Problems on Common Transversals and Submodular Functions from Welsh’s 1976 Text Matroid Theory
Author:
Jonathan David FarleyDepartment of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, United States of America

Search for other papers by Jonathan David Farley in
Current site
Google Scholar
PubMedClose
View More View Less
Pages:
8–16
Online Publication Date:
14 Oct 2022
Publication Date:
19 May 2023
Article Category:
Research Article
DOI:
https://doi.org/10.1556/314.2022.00023
Keywords:
Independence space; independence structure; matroid; submodular; (common) transversal
Open access

Problem 2 of Welsh’s 1976 text Matroid Theory, asking for criteria telling when two families of sets have a common transversal, is solved.

Another unsolved problem in the text Matroid Theory, on whether the “join” of two non-decreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.

  • [1]

    Aharoni, R., Nash-Williams, C. St. J. A. and Shelah, S. A General Criterion for the Existence of Transversals. Proceedings of the London Mathematical Society, 47 (1983), 43–68.

    • Search Google Scholar
    • Export Citation
  • [2]

    Barnard, A. D. Obituary: Aubrey William Ingleton (1920–2000). Bulletin of the London Mathematical Society, 36 (2004), 119–129.

  • [3]

    Brualdi, R. A. Common Transversals and Strong Exchange Systems. Journal of Combinatorial Theory, 8 (1970), 307–329.

  • [4]

    Brualdi, R. A. and Scrimger, E. B. Exchange Systems, Matchings, and Transversals. Journal of Combinatorial Theory, 5 (1968), 244–257.

  • [5]

    Edmonds, J. and Fulkerson, D. R. Transversals and Matroid Partition. Memorandum RM-4608-PR, The RAND Corporation (May 1965).

  • [6]

    Ford, L. R. Jr and Fulkerson, D. R. Network Flow and Systems of Representatives. Canadian Journal of Mathematics, 10 (1958), 78–84.

  • [7]

    Hall, P. On Representatives of Subsets. Journal of the London Mathematical Society, 10 (1935), 26–30.

  • [8]

    McDiarmid, C. A Note on a Theorem of R. Rado on Independent Transversals. Bulletin of the London Mathematical Society, 5 (1973), 315–316.

    • Search Google Scholar
    • Export Citation
  • [9]

    Milner, E. C. Independent Transversals for Countable Set Systems. Journal of Combinatorial Theory (A), 19 (1975), 125–133.

  • [10]

    Mirsky, L. Transversal Theory: An Account of Some Aspects of Combinatorial Mathematics. Academic Press, New York City, 1971.

  • [11]

    Mathematical Reviews MR0427112 (55 #148).

  • [12]

    Oxley, J. The Contributions of Dominic Welsh to Matroid Theory. In Geo_rey Grimmett and Colin McDiarmid (eds.), Combinatorics, Complexity, and Chance. A Tribute to Dominic Welsh. Oxford University Press, Oxford, 2007, 234–259.

    • Search Google Scholar
    • Export Citation
  • [13]

    Perfect, H. Remark on a Criterion for Common Transversals. Glasgow Mathematical Journal, 10 (1969), 66–67.

  • [14]

    Pym, J. S. and Perfect, H. Submodular Functions and Independence Structures. Journal of Mathematical Analysis and Applications, 30 (1970), 1–31.

    • Search Google Scholar
    • Export Citation
  • [15]

    Steffens, K. Ergebnisse aus der Transversalentheorie. II. Journal of Combinatorial Theory (A), 20 (1976), 202–229.

  • [16]

    University of Waterloo, Citation for Dominic Welsh (2006). https://uwaterloo.ca/combinatoricsand-optimization/sites/ca.combinatorics-and-optimization/_les/uploads/_les/dom-award-1.pdf

    • Search Google Scholar
    • Export Citation
  • [17]

    Welsh, D. J. A. Matroid Theory. Academic Press, Inc., New York City, 1976.

  • Collapse
  • Expand
Cover Mathematica Pannonica
Mathematica Pannonica
Print ISSN:
0865-2090
Online ISSN:
2786-0752
The Instruction for Authors is available in PDF format. Please, download the file from HERE.
Please, download the LaTeX template from HERE.

Editor in Chief: László TÓTH (University of Pécs)

Honorary Editors in Chief:

  • † István GYŐRI (University of Pannonia, Veszprém)
  • János PINTZ (Rényi Institute of Mathematics)
  • Ferenc SCHIPP (Eötvös University Budapest and University of Pécs)
  • Sándor SZABÓ (University of Pécs)
     

Deputy Editors in Chief:

  • Erhard AICHINGER (JKU Linz)
  • Ferenc HARTUNG (University of Pannonia, Veszprém)
  • Ferenc WEISZ (Eötvös University, Budapest)

Editorial Board

  • György DÓSA (University of Pannonia, Veszprém)
  • István BERKES (Rényi Institute of Mathematics)
  • Károly BEZDEK (University of Calgary)
  • Balázs KIRÁLY – Managing Editor (University of Pécs)
  • Vedran KRCADINAC (University of Zagreb) 
  • Željka MILIN ŠIPUŠ (University of Zagreb)
  • Margit PAP (University of Pécs)
  • Mihály PITUK (University of Pannonia, Veszprém)
  • Jörg THUSWALDNER (Montanuniversität Leoben)
  • Zsolt TUZA (University of Pannonia, Veszprém)

Advisory Board

  • Szilárd RÉVÉSZ (Rényi Institute of Mathematics)  - Chair
  • Gabriella BÖHM (Akadémiai Kiadó, Budapest)
  • György GÁT (University of Debrecen)

University of Pécs,
Faculty of Sciences,
Institute of Mathematics and Informatics
Department of Mathematics
7624 Pécs, Ifjúság útja 6., HUNGARY
(36) 72-503-600 / 4179
ltoth@gamma.ttk.pte.hu

  • Mathematical Reviews
  • Zentralblatt
  • DOAJ

Publication Model Gold Open Access
Submission Fee none
Article Processing Charge 0 EUR/article (temporarily)
Subscription Information Gold Open Access

Mathematica Pannonica
Language English
Size A4
Year of
Foundation		 1990
Volumes
per Year		 1
Issues
per Year		 2
Founder Akadémiai Kiadó
Founder's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Publisher Akadémiai Kiadó
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 2786-0752 (Online)
ISSN 0865-2090 (Print)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Dec 2022 0 18 11
Jan 2023 0 25 8
Feb 2023 0 16 17
Mar 2023 0 8 7
Apr 2023 0 16 13
May 2023 0 26 13
Jun 2023 0 0 0