DOCSLIB.ORG

Modularity and Structure in Matroids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modularity and Structure in Matroids Modularity and Structure in Matroids by Rohan Kapadia A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2013 c Rohan Kapadia 2013 AUTHOR'S DECLARATION I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract This thesis concerns sufficient conditions for a matroid to admit one of two types of structural characterization: a representation over a finite field or a description as a frame matroid. We call a restriction N of a matroid M modular if, for every flat F of M, rM (F ) + r(N) = rM (F \ E(N)) + rM (F [ E(N)): A consequence of a theorem of Seymour is that any 3-connected matroid with a modular U2;3-restriction is binary. We extend this fact to arbitrary finite fields, showing that if N is a modular rank-3 restriction of a vertically 4-connected matroid M, then any representation of N over a finite field extends to a representation of M. We also look at a more general notion of modularity that applies to minors of a matroid, and use it to present conditions for a matroid with a large projective geometry minor to be representable over a finite field. In particular, we show that a 3-connected, representable matroid with a sufficiently large projective geometry over a finite field GF(q) as a minor is either representable over GF(q) or has a U2;q2+1-minor. A second result of Seymour is that any vertically 4-connected matroid with a modular M(K4)-restriction is graphic. Geelen, Gerards, and Whittle partially generalized this from M(K4) to larger frame matroids, showing that any vertically 5-connected, representable matroid with a rank-4 Dowling geometry as a modular restriction is a frame matroid. As with projective geometries, we prove a version of this result for matroids with large Dowling geometries as minors, providing conditions which imply that they are frame matroids. iii Acknowledgements Above all, I would like to thank Jim Geelen for his guidance and inspiration. It has been a great experience being his student and it is an honour to have been able to learn so much from him. I thank my examining committee members, Rahim Moosa, James Oxley, Bruce Richter, and David Wagner, for their care in reading this thesis and their helpful comments. I would also like to thank everyone who has been at the Department of Combinatorics and Optimization during my time here for making it such a great place to be. This work was partially supported by a scholarship from the Natural Sciences and Engineering Research Council of Canada. v Table of Contents Abstract iii Acknowledgementsv Table of Contents vii 1 Introduction1 1.1 Matroids..............................2 1.1.1 Minors and duality....................3 1.1.2 Projective geometries...................4 1.1.3 Graphic matroids.....................5 1.1.4 Frame and Dowling matroids..............6 1.1.5 Connectivity........................8 1.2 Varieties of matroids.......................9 1.3 Three in a circuit and two disjoint rooted paths........ 10 1.4 Modular restrictions........................ 11 1.5 Conditions for representation over a field............ 14 1.6 Conditions for a frame representation.............. 16 1.7 Excluded minors of varieties.................... 17 1.8 Growth rates of minor-closed classes.............. 19 2 Modular planes 23 2.1 Modular sums........................... 25 2.2 General modular restrictions.................... 27 2.3 Duality............................... 29 2.4 Finding a deletion pair...................... 35 2.5 Stabilizers............................. 52 2.6 Finding distinguishing sets.................... 53 2.7 Connectivity............................. 61 2.8 Vertically 4-connected matroids.................. 71 vii 2.9 Excluded minors......................... 74 3 Projective geometries 77 3.1 Non-representable matroids................... 78 3.2 Representation over a subfield................... 81 3.3 The main theorem........................ 84 3.4 Growth rates............................ 87 4 Dowling geometries 89 4.1 Represented matroids........................ 91 4.2 Patchworks............................. 91 4.3 Group-labelled graphs...................... 94 4.4 A non-abelian group....................... 95 4.5 Coextensions of Dowling matroids................. 97 4.6 Vertex- and edge-labelled graphs................ 109 4.7 Unique maximal skeletons..................... 111 4.8 Tangles.............................. 119 4.9 Matroids with a Dowling geometry minor............ 122 4.10 Three elements in a circuit.................... 133 References 137 Glossary of Notation 143 Index 145 viii Chapter 1 Introduction For many types of combinatorial objects, we can often prove that each object either admits a well-understood structural description or has a small obstruction to it. A famous example is Kuratowski's theorem that any graph is either planar or has one of two specific graphs, K5 and K3;3, as a minor [34]. A remarkable extension of this fact is Robertson and Seymour's graph minor theorem, which says that every class of graphs closed under the minor relation has a finite set of excluded minors, or minor-minimal graphs not in the class [43]. Moving to the realm of matroids, Tutte showed that every matroid is either graphic or has one of five particular non-graphic matroids as a minor [53]. Graphicness is one of the two most common types of structural description we encounter for matroids; the other is that of representability over a fixed finite field. Tutte showed that there is a unique excluded minor for the class of binary matroids [51], and there are four and seven excluded minors, respectively, for the matroids representable over the fields of order three and four [4, 48, 14]. For each larger finite field F, Geelen, Gerards, and Whittle have recently announced a proof of Rota's Conjecture [44], which asserts that there are finitely many excluded minors for the class of F-representable matroids. In this thesis, we consider a variant of these excluded-minor questions for matroids, exhibiting sufficient conditions for a matroid to be representable over a finite field F. When a matroid M has an F-representable minor with particular properties, we show that either M itself is F-representable or that it has a specific small minor obstruction. An example is the following consequence of one of our main theorems. Theorem 1.0.1. Any vertically 4-connected matroid with PG(2; q) as a re- striction is either GF(q)-representable or has a U2;q2+1-minor. In addition to representability, we will also investigate conditions under 1 2 CHAPTER 1. INTRODUCTION which a matroid belongs to a certain class of `graph-like' matroids called the frame matroids over a field, which are those that have a representation with at most two non-zero entries per column. A special case of one of our theorems is that if a highly connected representable matroid M has a particular large frame matroid called a Dowling geometry as a minor, then M is either also a frame matroid or has a small minor obstruction to being a frame matroid. In the next section, we summarize the concepts of matroid theory that we refer to in this chapter. Next, we define varieties of matroids, and then we present two striking theorems of Seymour that give sufficient conditions for a matroid to be binary and to be graphic. We then outline the main results of the thesis, which mainly consist of extensions of these two theorems in various directions. Finally, we discuss two topics to which our results relate: excluded minors for varieties of matroids and growth-rate functions of minor-closed classes. 1.1 Matroids Whitney introduced matroids as a way to capture the linear dependence proper- ties of finite subsets of a finite-dimensional vector space [55]. A comprehensive reference on matroid theory can be found in Oxley [37]. We let F be a field, E a finite set, and A 2 Fk×E a k × jEj matrix whose columns are indexed by the elements of E. For each X ⊆ E, we denote by AjX the restriction of A to the set of columns X. The rank of the matrix AjX, rank(AjX), is equal to the dimension of the subspace of Fk spanned by the columns indexed by X. We abbreviate rank(AjX) as r(X); this function always satisfies three properties: (R1)0 ≤ r(X) ≤ jXj, for all X ⊆ E, (R2) r(X) ≤ r(Y ), for all X ⊆ Y ⊆ E, and (R3) r(X) + r(Y ) ≥ r(X \ Y ) + r(X [ Y ), for all X; Y ⊆ E. We take these three properties as axioms to define a more general class of objects. A matroid is a pair M = (E; r) consisting of a finite set E and a function r : 2E ! Z that satisfies (R1)-(R3). We call E the ground set of M and r(X) the rank of a set X ⊆ E(M), and to avoid ambiguity we will always write E(M) and rM for the ground set and rank function of a matroid M. The rank of a matroid M, denoted r(M), is equal to rM (E(M)). An F-representation of a matroid M is an F-matrix A with columns indexed by E(M) such that for all X ⊆ E(M), rM (X) is actually the rank of the 1.1. MATROIDS 3 matrix AjX. We call a matroid representable over F or F-representable if it has an F-representation, and representable if it has a representation over some field. Conversely, when A is an F-matrix, we write MF(A) for the matroid with A as an F-representation.
Load more

Recommended publications
  • On Excluded Minors and Biased Graph Representations of Frame Matroids
    On excluded minors and biased graph representations of frame matroids by Daryl Funk M.Sc., University of Victoria, 2009 B.Sc., Simon Fraser University, 1992 Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science c Daryl Funk 2015 SIMON FRASER UNIVERSITY Spring 2015 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for \Fair Dealing." Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Daryl Funk Degree: Doctor of Philosophy (Mathematics) Title of Thesis: On excluded minors and biased graph representations of frame matroids Examining Committee: Dr. Jonathan Jedwab, Chair Professor, Department of Mathematics Dr. Matthew DeVos Senior Supervisor Associate Professor, Department of Mathematics Dr. Luis Goddyn Co-Supervisor Professor, Department of Mathematics Dr. Bojan Mohar Internal Examiner Professor, Department of Mathematics Dr. Daniel Slilaty External Examiner Professor, Department of Department of Mathematics and Statistics Wright State University Date Defended: 8 January 2015 ii Partial Copyright Licence iii ABSTRACT A biased graph is a graph in which every cycle has been given a bias, either balanced or un- balanced. Biased graphs provide representations for an important class of matroids, the frame matroids. As with graphs, we may take minors of biased graphs and of matroids, and a family of biased graphs or matroids is minor-closed if it contains every minor of every member of the family.
    [Show full text]
  • Projective Planarity of Matroids of 3-Nets and Biased Graphs
    AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 76(2) (2020), Pages 299–338 Projective planarity of matroids of 3-nets and biased graphs Rigoberto Florez´ ∗ Deptartment of Mathematical Sciences, The Citadel Charleston, South Carolina 29409 U.S.A. [email protected] Thomas Zaslavsky† Department of Mathematical Sciences, Binghamton University Binghamton, New York 13902-6000 U.S.A. [email protected] Abstract A biased graph is a graph with a class of selected circles (“cycles”, “cir- cuits”), called “balanced”, such that no theta subgraph contains exactly two balanced circles. A 3-node biased graph is equivalent to an abstract partial 3-net. We work in terms of a special kind of 3-node biased graph called a biased expansion of a triangle. Our results apply to all finite 3-node biased graphs because, as we prove, every such biased graph is a subgraph of a finite biased expansion of a triangle. A biased expansion of a triangle is equivalent to a 3-net, which, in turn, is equivalent to an isostrophe class of quasigroups. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this paper we establish criteria for the remaining case, that is, embeddability in an arbitrary projective plane that is not necessarily Desarguesian.
    [Show full text]
  • Templates for Representable Matroids Kevin Manuel Grace Louisiana State University and Agricultural and Mechanical College, [email protected]
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 6-12-2018 Templates for Representable Matroids Kevin Manuel Grace Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Grace, Kevin Manuel, "Templates for Representable Matroids" (2018). LSU Doctoral Dissertations. 4610. https://digitalcommons.lsu.edu/gradschool_dissertations/4610 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. TEMPLATES FOR REPRESENTABLE MATROIDS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Kevin Manuel Grace B.S., Pensacola Christian College, 2006 M.S., University of South Alabama, 2010 August 2018 \It is the glory of God to conceal a thing: but the honour of kings is to search out a matter." Proverbs 25:2 ii Acknowledgments It is a profound understatement to say that this dissertation would not have been possible without my advisor, Professor Stefan van Zwam. He has had a tremendous influence on every aspect of this dissertation. I would especially like to thank him for his guidance on presenting my research and for his help with the programming that was necessary. It was an honor to be his first graduate student.
    [Show full text]
  • On the Excluded Minors for Matroids of Branch-Width Three
    On the Excluded Minors for Matroids of Branch-Width Three Petr Hlinˇen´y∗ School of Mathematical and Computing Sciences, Victoria University, P.O. Box 600, Wellington, New Zealand; and Institute for Theoretical Computer Science†(ITI MFF), Charles University, Malostransk´en´am. 25, 118 00 Praha 1, Czech Republic. [email protected] Submitted: March 19, 2002; Accepted: July 23, 2002. Abstract Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [R. Hall, J. Oxley, C. Semple, G. Whittle, On Matroids of Branch-Width Three, submitted 2001] that if M is an excluded minor for matroids of branch-width 3, then M has at most 14 elements. We show that there are exactly 10 such binary matroids M (7 of which are regular), proving a conjecture formulated by Dharmatilake in 1994. We also construct numbers of such ternary and quaternary matroids M, and provide a simple practical algorithm for finding a width-3 branch-decomposition of a matroid. The arguments in our paper are computer-assisted | we use a program Macek [P. Hlinˇen´y, The Macek Program, http://www.mcs.vuw.ac.nz/research/macek, 2002] for structural computations with represented matroids. Unfortunately, it seems to be infeasible to search through all matroids on at most 14 elements. Keywords: representable matroid, minor, branch-width. MR Subject Classifications: 05B35, 05C83 ∗The research was supported by a New Zealand Marsden Fund research grant to Geoff Whittle. y(Supported by the Ministry of Education of Czech Republic as project LN00A056.) the electronic journal of combinatorics 9 (2002), #R32 1 1 Introduction We assume that the reader is familiar with basic terms of graph theory.
    [Show full text]
  • Well-Quasi-Ordering of Matrices Under Schur Complement And
    Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs Mamadou Moustapha Kanté Clermont-Université, Université Blaise Pascal, LIMOS, CNRS Complexe Scientifique des Cézeaux 63173 Aubiére Cedex, France [email protected] Abstract In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matri- ces, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F, any infinite se- quence M1, M2,... of (skew) symmetric matrices over F of bounded F-rank-width has a pair i < j, such that Mi is isomorphic to a principal submatrix of a principal pivot transform of Mj. We generalise this result to σ-symmetric matrices introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs, arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of σ-symmetric matrices. As a by- product, we obtain that for every infinite sequence G1, G2,... of directed graphs of bounded rank-width there exist a pair i < j such that Gi is a pivot-minor of Gj. Another consequence is that non-singular principal submatrices of a σ-symmetric matrix form a delta-matroid. We extend in this way the notion of representability of delta-matroids by Bouchet. Key words: rank-width; sigma-symmetry; edge-coloured graph; well-quasi-ordering; principal pivot transform; pivot-minor. arXiv:1102.2134v3 [math.CO] 8 Jul 2014 1 Introduction Clique-width [6] is a graph complexity measure that emerges in the works by Courcelle et al. (see for instance the book [7]). It extends tree-width [21] in the sense that graph classes of bounded tree-width have bounded clique- width, but the converse is false (distance hereditary graphs have clique-width at most 3 and unbounded tree-width).
    [Show full text]
  • Graphs of Bounded Rank-Width
    Graphs of Bounded Rank-width Sang-il Oum A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Program in Applied and Computational Mathematics May 2005 c Copyright by Sang-il Oum, 2005. All Rights Reserved iii Abstract We define rank-width of graphs to investigate clique-width. Rank-width is a com- plexity measure of decomposing a graph in a kind of tree-structure, called a rank- decomposition. We show that graphs have bounded rank-width if and only if they have bounded clique-width. It is unknown how to recognize graphs of clique-width at most k for fixed k > 3 in polynomial time. However, we find an algorithm recognizing graphs of rank-width at most k, by combining following three ingredients. First, we construct a polynomial-time algorithm, for fixed k, that confirms rank- width is larger than k or outputs a rank-decomposition of width at most f(k) for some function f. It was known that many hard graph problems have polynomial-time algo- rithms for graphs of bounded clique-width, however, requiring a given decomposition corresponding to clique-width (k-expression); we remove this requirement. Second, we define graph vertex-minors which generalizes matroid minors, and prove that if {G1,G2,...} is an infinite sequence of graphs of bounded rank-width, then there exist i < j such that Gi is isomorphic to a vertex-minor of Gj. Conse- quently there is a finite list Ck of graphs such that a graph has rank-width at most k if and only if none of its vertex-minors are isomorphic to a graph in Ck.
    [Show full text]
  • Varieties of Combinatorial Geometries by J
    transactions of the american mathematical society Volume 271, Number 2, June 1982 VARIETIES OF COMBINATORIAL GEOMETRIES BY J. KAHN1 AND J. P. S. KUNG Abstract. A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class 'S of geometries is a sequence (T„ ) of geometries in ?T with rank Tn = n, and satisfying the universal property: if G is a geometry in 5" of rank n, then G is a subgeometry of T„. A variety of geometries is a hereditary class with a sequence of universal models. We prove that, apart from two degenerate cases, the only varieties of combina- torial geometries are ( 1) the variety of free geometries, (2) the variety of geometries coordinatizable over a fixed finite field, and (3) the variety of voltage-graphic geometries with voltages in a fixed finite group. 1. Introduction. The notion of free objects in a variety is one of the most important and pervasive ideas in universal algebra. In this paper, we investigate how that notion can be interpreted in the context of the theory of combinatorial geometries (or matroids). No detailed knowledge of universal algebra is required for reading this paper. However, we do assume familiarity with the basic concepts of the theory of combinatorial geometries [2, 4 and 13]. To fix our terminology, let G be a finite geometric lattice. Its maximum and minimum are denoted by 1 and 0. Let S be the set of points (or atoms) in G.
    [Show full text]
  • Matroid Pathwidth and Code Trellis Complexity∗ 1
    MATROID PATHWIDTH AND CODE TRELLIS COMPLEXITY∗ NAVIN KASHYAP† Abstract. We relate the notion of matroid pathwidth to the minimum trellis state-complexity (which we term trellis-width) of a linear code, and to the pathwidth of a graph. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of determining the pathwidth of a representable matroid is NP-hard. Consequently, the problem of computing the trellis-width of a linear code is also NP-hard. For a finite field F, we also consider the class of F-representable matroids of pathwidth at most w, and correspondingly, the family of linear codes over F with trellis-width at most w. These are easily seen to be minor-closed. Since these matroids (and codes) have branchwidth at most w, a result of Geelen and Whittle shows that such matroids (and the corresponding codes) are characterized by finitely many excluded minors. We provide the complete list of excluded minors for w = 1, and give a partial list for w = 2. Key words. Matroids, pathwidth, linear codes, trellis complexity, NP-hard. AMS subject classifications. 05B35, 94B05 1. Introduction. The notion of pathwidth of a matroid has received some recent attention in the matroid theory literature [7], [9]. This notion has long been studied in the coding theory literature, where it is used as a measure of trellis complexity of a linear code [14], [5], [17]. However, there appears to be no standard coding-theoretic nomenclature for this notion. It has been called the state complexity of a code in [10], but the use of this term there conflicts slightly with its use in [17].
    [Show full text]
  • A Characterisation of Jointless Dowling Geometries Irasema Sarmiento* Mathematical Institute, Oxjord Untver.Vity, 24-29 St
    DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 197/198 ( 1999 ) 713--731 A characterisation of jointless Dowling geometries Irasema Sarmiento* Mathematical Institute, Oxjord Untver.vity, 24-29 St. Giles', O.'@~rd OX1 3LB. UK Received 9 July 1997; revised 18 December 1997; accepted 3 August 1998 Abstract We use statistics of flats of small rank in order to characterise the jointless Dowling geometries defined by groups of order exceeding three and having rank greater than 3. In particular, we show that if the Tutte polynomial of a matroid is identical to the Tutte polynomial of a jointless Dowling geometry, then the matroid is indeed a jointless Dowling geometry. For rank 3 (and groups of order exceeding 3) this holds only if the order of the group is even. @ 1999 Elsevier Science B.V. All rights reserved 1. Introduction Characterising classes of matroids by their Tutte polynomials is particularly inter- esting due to the large number of applications of Tutte polynomials [7]. Only a few classes of matroids have been shown to be characterised by their Tutte polynomials. Examples of such classes of matroids are projective and affine geometries and Dowling lattices [4]. We will prove here that jointless Dowling lattices are also characterised by their Tutte polynomials. Dowling lattices are group-theoretic generalisations of partition lattices [108]. They have, as well, an interpretation as matroids of gain graphs [14], and are, together with representable matroids, the only non-degenerate varieties [10]. Similarities between projective geometries and Dowling lattices have been studied in [1,2]. Dowling lattices form an infinite family of supersolvable tangential blocks [13].
    [Show full text]
  • SKEW PARTIAL FIELDS, MULTILINEAR REPRESENTATIONS of MATROIDS, and a MATRIX TREE THEOREM for Geoff Whittle, Whose Insights and Ki
    SKEW PARTIAL FIELDS, MULTILINEAR REPRESENTATIONS OF MATROIDS, AND A MATRIX TREE THEOREM R.A. PENDAVINGH AND S.H.M. VAN ZWAM ABSTRACT. We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such represen- tations extends Tutte’s definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte’s repre- sentability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the Matrix Tree theorem for this class. MSC subject classification: 05B35 Keywords: matroids, representations, skew partial fields, multilinear rep- resentations, matrix tree theorem For Geoff Whittle, whose insights and kindness have made working in ma- troid theory even more rewarding. 1. INTRODUCTION A matrix with entries in R is totally unimodular if the determinant of each square submatrix is in 1, 0, 1 . A matroid is regular if it can be represented by a totally unimodular{− matrix.g Regular matroids are well-studied objects with many attractive properties. For instance, a binary matroid is either regular, and therefore representable over every field, or it is representable only over fields of characteristic 2. Whittle proved a similar, but more complicated, classification of the repre- sentability of ternary matroids [24, 25].
    [Show full text]
  • Secret-Sharing Matroids Need Not Be Algebraic
    Secret-Sharing Matroids need not be Algebraic Aner Ben-Efraim∗ September 27, 2018 Abstract We combine some known results and techniques with new ones to show that there exists a non- algebraic, multi-linear matroid. This answers an open question by Mat´uˇs(Discrete Mathematics 1999), and an open question by Pendavingh and van Zwam (Advances in Applied Mathematics 2013). The proof is constructive and the matroid is explicitly given. Keywords: Algebraic Matroids, Multi-Linear Matroids. 1 Introduction The aim of this paper is to compare two natural extensions of linear representations of matroids: multi- linear representations and algebraic representations. Specifically, we show that there exists a non-algebraic, multi-linearly representable matroid. Linear matroids. The beginning of matroid theory dates back to the work of Whitney in 1935. Whitney defined the matroid axioms as an abstraction of the linear dependence properties of a set of vectors. It is therefore not surprising that the class of linear matroids, i.e., matroids that originate from linear dependence, is probably the most well studied class of matroids. The class of linear matroids has many desirable properties, such as closure to minors and duality. Quite early in the study of matroids, MacLane proved that not all matroids are linear. The smallest non-linear matroid was presented by V´amos [20] (the V´amos matroid). Since linear algebra can also be done over division rings, and the linear dependence over division rings arXiv:1403.6363v1 [math.CO] 25 Mar 2014 clearly satisfies the matroid axioms, the class of matroids representable over division rings is probably the most natural extension of linear matroids.
    [Show full text]
  • Binary Matroid Whose Bases Form a Graphic Delta-Matroid
    Characterizing matroids whose bases form graphic delta-matroids Duksang Lee∗2,1 and Sang-il Oum∗1,2 1Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea 2Department of Mathematical Sciences, KAIST, Daejeon, South Korea Email: [email protected], [email protected] September 1, 2021 Abstract We introduce delta-graphic matroids, which are matroids whose bases form graphic delta- matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural charac- terization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most 48 elements. 1 Introduction Bouchet [2] introduced delta-matroids which are set systems admitting a certain exchange axiom, generalizing matroids. Oum [18] introduced graphic delta-matroids as minors of binary delta-matroids having line graphs as their fundamental graphs and proved that bases of graphic matroids form graphic delta-matroids. We introduce delta-graphic matroids as matroids whose bases form a graphic delta- matroid. Since the class of delta-graphic matroids is closed under taking dual and minors, it contains both graphic matroids and cographic matroids. Alternatively one can define delta-graphic matroids as binary matroids whose fundamental graphs are pivot-minors of line graphs. See Section 2 for the definition of pivot-minors. Our first theorem provides a structural characterization of delta-graphic matroids. A wheel graph W is a graph having a vertex s adjacent to all other vertices such that W \s is a cycle.
    [Show full text]