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memo_has_moved_text();The Internally $4$-Connected Binary Matroids With No $M(K_{3,3})$-Minor.

About this Title

Dillon Mayhew, School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, P.O. BOX 600, Wellington, New Zealand., Gordon Royle, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley 6009, Western Australia. and Geoff Whittle, School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, P.O. BOX 600, Wellington, New Zealand.

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 208, Number 981
ISBNs: 978-0-8218-4826-5 (print); 978-1-4704-0595-3 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00600-9
Published electronically: June 8, 2010
MathSciNet review: 2742785
Keywords:Binary matroids, excluded minors
MSC: Primary 05B35

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Table of Contents

Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. Möbius matroids
• Chapter 4. From internal to vertical connectivity
• Chapter 5. An $R_{12}$-type matroid
• Chapter 6. A connectivity lemma
• Chapter 7. Proof of the main result
• Appendix A. Case-checking
• Appendix B. Sporadic matroids
• Appendix C. Allowable triangles

Abstract

We give a characterization of the internally $4$-connected binary matroids that have no minor isomorphic to $M(K_{3,3})$. Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Möbius ladder, or is isomorphic to one of eighteen sporadic matroids.

References [Enhancements On Off] (What's this?)

• [AO93] Safwan Akkari and James Oxley, Some local extremal connectivity results for matroids, Combin. Probab. Comput. 2 (1993), no. 4, 367-384. MR 1264712 (94k:05048)
• [BCG98] André Bouchet, W. H. Cunningham, and J. F. Geelen, Principally unimodular skew-symmetric matrices, Combinatorica 18 (1998), no. 4, 461-486. MR 1722253 (2001a:05097)
• [Bix82] Robert E. Bixby, A simple theorem on $3$-connectivity, Linear Algebra Appl. 45 (1982), 123-126. MR 660982 (84j:05037)
• [Bry75] Tom Brylawski, Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203 (1975), 1-44. MR 0357163 (50:9631)
• [GGK00] J. F. Geelen, A. M. H. Gerards, and A. Kapoor, The excluded minors for ${\rm GF}(4)$-representable matroids, J. Combin. Theory Ser. B 79 (2000), no. 2, 247-299. MR 1769191 (2001f:05039)
• [GZ06] Jim Geelen and Xiangqian Zhou, A splitter theorem for internally $4$-connected binary matroids, SIAM J. Discrete Math. 20 (2006), no. 3, 578-587 (electronic). MR 2272214 (2007j:05039)
• [Hal43] Dick Wick Hall, A note on primitive skew curves, Bull. Amer. Math. Soc. 49 (1943), 935-936. MR 0009442 (5:151b)
• [Hal02] Matthew Halfan, Matroid decomposition, Master's essay, University of Waterloo, 2002.
• [Kin97] S. R. Kingan, A generalization of a graph result of D. W. Hall, Discrete Math. 173 (1997), no. 1-3, 129-135. MR 1468845 (98c:05039)
• [KL02] S. R. Kingan and Manoel Lemos, Almost-graphic matroids, Adv. in Appl. Math. 28 (2002), no. 3-4, 438-477, Special issue in memory of Rodica Simion. MR 1900004 (2004c:05047)
• [Kun86] Joseph P. S. Kung, Growth rates and critical exponents of classes of binary combinatorial geometries, Trans. Amer. Math. Soc. 293 (1986), no. 2, 837-859. MR 816330 (87f:05043)
• [MRW] Dillon Mayhew, Gordon Royle, and Geoff Whittle, Excluding Kuratowski graphs and their duals from binary matroids, submitted. Available at arXiv:0902.0198v1 [math.CO].
• [OSV00] James Oxley, Charles Semple, and Dirk Vertigan, Generalized $\Delta\textrm{-}Y$ exchange and $k$-regular matroids, J. Combin. Theory Ser. B 79 (2000), no. 1, 1-65. MR 1757022 (2001d:05035)
• [OSW04] James Oxley, Charles Semple, and Geoff Whittle, The structure of the $3$-separations of $3$-connected matroids, J. Combin. Theory Ser. B 92 (2004), no. 2, 257-293. MR 2099144 (2005h:05038)
• [Oxl87] James G. Oxley, On nonbinary $3$-connected matroids, Trans. Amer. Math. Soc. 300 (1987), no. 2, 663-679. MR 876471 (88c:05046)
• [Oxl92] -, Matroid theory, Oxford University Press, New York, 1992. MR 1207587 (94d:05033)
• [QZ04] Hongxun Qin and Xiangqian Zhou, The class of binary matroids with no $M(K_{3,3})$-, $M^*(K_{3,3})$-, $M(K_5)$- or $M^*(K_5)$-minor, J. Combin. Theory Ser. B 90 (2004), no. 1, 173-184. MR 2041325 (2005a:05058)
• [Sey80] P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), no. 3, 305-359. MR 579077 (82j:05046)
• [Tru86] K. Truemper, A decomposition theory for matroids. III. Decomposition conditions, J. Combin. Theory Ser. B 41 (1986), no. 3, 275-305. MR 864578 (88b:05045)
• [Tut58] W. T. Tutte, A homotopy theorem for matroids. I, II, Trans. Amer. Math. Soc. 88 (1958), 144-174. MR 0101526 (21:336)
• [Wag37] K. Wagner, Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), no. 1, 570-590. MR 1513158
• [WW80] P. N. Walton and D. J. A. Welsh, On the chromatic number of binary matroids, Mathematika 27 (1980), no. 1, 1-9. MR 581990 (81m:05060)
• [Zho04] Xiangqian Zhou, On internally $4$-connected non-regular binary matroids, J. Combin. Theory Ser. B 91 (2004), no. 2, 327-343. MR 2064874 (2005e:05026)
• [Zho08] -, A note on binary matroid with no ${M}({K}_{3,3})$-minor, J. Combin. Theory Ser. B 98 (2008), no. 1, 235-238. MR 2368034 (2008j:05087)

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