Dirk Vertigan

Louisiana State University, Baton Rouge, Louisiana, United States

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Publications (32)12.67 Total impact

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    ABSTRACT: We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices. Comment: 20 pages. No figures. TeX. Submitted
    12/2009;
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    Dirk Vertigan
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    ABSTRACT: We consider what the implications would be if there were a discrete fundamental model of physics based on locally-finite self-interacting information, in which there is no presumption of the familiar space and laws of physics, but from which such space and laws can nevertheless be shown to be able to emerge stably from such a fundamental model. We argue that if there is such a model, then the familiar laws of physics, including Standard Model constants, etc., must be encodable by a finite quantity C, called the complexity, of self-interacting information I, called a Space-Cell. Copies of Space-Cell I must be distributed throughout space, at a roughly constant and near-Planck density, and copies must be created or destroyed as space expands or contracts. We then argue that each Space-Cell is a self-replicator that can duplicate in times ranging from as fast as near-Planck-times to as slow as Cosmological-Constant-time which is 10^{61} Planck-times. From standard considerations of computation, we argue this slowest duplication rate just requires that 10^{61} is less than about 2^C, the number of length-C binary strings, hence requiring only the modest complexity C at least 203, and at most a few thousand. We claim this provides a reasonable explanation for a dimensionless constant being as large as 10^{61}, and hence for the Cosmological Constant being a tiny positive 10^{-122}. We also discuss a separate conjecture on entropy flow in Hole-Bang Transitions. We then present Cosmological Natural Selection II.
    06/2007;
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    ABSTRACT: A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K_{4,k} with a complete graph on the vertices of degree k, the k-partition triple fan with a complete graph on the vertices of degree k, the k-spoke double wheel, the k-spoke double wheel with axle, the (2k+1)-rung Mobius zigzag ladder, the (2k)-rung zigzag ladder, or K_k. We also find the unavoidable parallel minors of 1-, 2-, and 3-connected graphs.
    Journal of Graph Theory 07/2006; · 0.63 Impact Factor
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    ABSTRACT: An algorithm is given for computing the weights of extensions of BCH codes embedded in semigroup rings as ideals. The algorithm relies on a more general technical result of independent interest.
    Agronomy Journal 01/2006; 73(3):317-329. · 0.46 Impact Factor
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    ABSTRACT: Let G be a directed graph embedded in a surface. A map : E(G) ! R is a tension if for every circuit C G, the sum of on the forward edges of C is equal to the sum of on the backward edges of C. If this condition is satised for every circuit of G which is a contractible curve in the surface, then is a local tension. If 1 j (e)j 1 holds for every e 2 E(G), we say that is a (local) -tension. We dene the circular chromatic number and the local circular chromatic number of G by c(G) = inff 2 R j G has an -tensiong and loc(G) = inff 2 R j G has a local -tensiong, respectively. The invariant c is a renemen t of the usual chromatic number, whereas loc is closely related to Tutte's o w index and Bouchet's bio w index of the surface dual G. From the denitions we have loc(G) c(G). The main result of this paper is a far reaching generalization of Tutte's coloring-o w duality in planar graphs. It is proved that for every surface X and every " > 0, there exists an integer M so that c(G) loc(G) + " holds for every graph embedded in X with edge-width at least M, where the edge-width is the length of a shortest noncontractible circuit in G. In 1996, Youngs discovered that every quadrangulation of the projective plane has chromatic number 2 or 4, but never 3. As an application of the main result we show that such 'bimodal' behavior can be observed in loc, and thus in c for two generic classes of embedded graphs: those that are triangulations and those whose face boundaries all have even length. In particular, if G is embedded in some surface with large edge-width and all its faces have even length 2r, then c(G) 2 (2; 2+ ")(( 2r r 1 ; 4). Similarly, if G is a triangulation with large edge-width, then c(G) 2 (3; 3+")((4; 5). It is also shown that there exist Eulerian triangulations of arbitrarily large edge-width on nonorientable surfaces whose circular chromatic number is equal to 5.
    Transactions of the American Mathematical Society 01/2005; 357(10). · 1.02 Impact Factor
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    ABSTRACT: Tutte proved that matroid is binary if and only if it does not contain a U2;4{ minor. This provides a short proof for non{GF (2){representability in that we can verify that a given minor is isomorphic to U2;4 in just a few rank evaluations. Using excludedminor characterizations, short proofs can also be given of non{representablity over GF(3) and over GF(4). For GF(5), it is not even known whether the set of excluded minors is nite. Nevertheless, we show here that if a matroid is not representable over GF(5), then this can be veried by a short proof. Here a short proof" is a proof whose length is bounded by some polynomial in the number of elements of the matroid. In contrast to these positive results, Seymour showed that we require exponentially many rank evaluations to prove GF(2){representability, and this is in fact the case for any eld. 1.
    J. Comb. Theory, Ser. B. 01/2004; 91:105-121.
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    ABSTRACT: This article proves the conjecture of Thomas that, for every graph G, there is an integer k such that every graph with no minor isomorphic to G has a 2-coloring of either its vertices or its edges where each color induces a graph of tree-width at most k. Some generalizations are also proved.
    Journal of Combinatorial Theory, Series B. 01/2004;
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    ABSTRACT: The paper presents several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. We show that every graph of bounded tree-width and bounded maximum degree admits such partitions. We also show that an arbitrary graph of maximum degree four has a vertex partition into two graphs, each of which has components on at most 57 vertices. Some generalizations of the last result are also discussed.
    Journal of Combinatorial Theory, Series B. 01/2003;
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    ABSTRACT: For each prime p, we construct an infinite antichain of matroids in which each matroid has characteristic set fpg. For p = 2, each of the matroids in our antichain is an excluded minor for the class of matroids representable over the rationals.
    Discrete Mathematics 01/2002; 242:175-185. · 0.58 Impact Factor
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    J. Comb. Theory, Ser. B. 01/2002; 84:130-179.
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    James Oxley, Charles Semple, Dirk Vertigan
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    ABSTRACT: This paper introduces a generalization of the matroid operation of Δ–Y exchange. This new operation, segment–cosegment exchange, replaces a coindependent set of k collinear points in a matroid by an independent set of k points that are collinear in the dual of the resulting matroid. The main theorem of the first half of the paper is that, for every field, or indeed partial field, F, the class of matroids representable over F is closed under segment–cosegment exchanges. It follows that, for all prime powers q, the set of excluded minors for GF(q)-representability has at least 2q−4 members. In the second half of the paper, the operation of segment–cosegment exchange is shown to play a fundamental role in an excluded-minor result for k-regular matroids, where such matroids generalize regular matroids and Whittle's near-regular matroids.
    Journal of Combinatorial Theory Series B 05/2000; · 0.85 Impact Factor
  • James Oxley, Dirk Vertigan
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    ABSTRACT: The classes of near-regular and 6 p 1{matroids arise in the study of matroids representable over GF (3) and other elds. For example, a matroid is representable over all elds except possibly GF (2) if and only if it is nearregular, and a matroid is representable over GF (3) and GF (4) if and only if it is a 6 p 1{matroid. This paper determines the maximum sizes of a simple rank{r near-regular and a simple rank{r 6 p 1{matroid and determines all such matroids having these sizes. 1.
    01/2000;
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    ABSTRACT: The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F -representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r⩾4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour's Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence.
    Journal of Combinatorial Theory Series B 01/2000; · 0.85 Impact Factor
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    ABSTRACT: LetFbe a field and letNbe a matroid in a class ofF-representable matroids that is closed under minors and the taking of duals. ThenNis anF-stabilizer for if every representation of a 3-connected member of is determined up to elementary row operations and column scaling by a representation of any one of itsN-minors. The study of stabilizers was initiated by Whittle. This paper extends that study by examining certain types of stabilizers and considering the connection with weak maps.The notion of a universal stabilizer is introduced to identify the underlying matroid structure that guarantees thatNwill be anF′-stabilizer for for every fieldF′ over which members of are representable. It is shown that, just as withF-stabilizers, one can establish whether or notNis a universal stabilizer for by an elementary finite check. IfNis a universal stabilizer for , we determine additional conditions onNand that ensure that ifNis not a strict rank-preserving weak-map image of any matroid in , then no connected matroid in with anN-minor is a strict rank-preserving weak-map image of any 3-connected matroid in .Applications of the theory are given for quaternary matroids. For example, it is shown thatU2, 5is a universal stabilizer for the class of quaternary matroids with noU3, 6-minor. Moreover, ifM1andM2are distinct quaternary matroids withU2, 5-minors but noU3, 6-minors andM1is connected whileM2is 3-connected, thenM1is not a rank-preserving weak-map image ofM2.
    Advances in Applied Mathematics 01/2000; · 0.84 Impact Factor
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    ABSTRACT: This paper strengthens the excluded-minor characterization of GF(4)-representable matroids. In particular, it is shown that there are only finitely many 3-connected matroids that are not GF(4)-representable and that have no U2, 6-, U4, 6-, P6-, F−7-, or (F−7)*-minors. Explicitly, these matroids are all minors of S(5, 6, 12) with rank and corank at least 4, and P″8, the matroid that can be obtained from S(5, 6, 12) by deleting two elements, contracting two elements, and then relaxing the only pair of disjoint circuit-hyperplanes.
    J. Comb. Theory, Ser. B. 01/2000; 80:57-68.
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    ABSTRACT: In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph. Some partial results towards this conjecture are presented. One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor. Several open questions are raised.
    Journal of Combinatorial Theory Series B 01/2000; · 0.85 Impact Factor
  • James G. Oxley, Charles Semple, Dirk L. Vertigan
    J. Comb. Theory, Ser. B. 01/2000; 79:1-65.
  • Combinatorica 01/1998; 18:1-12. · 0.56 Impact Factor
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    Dirk L. Vertigan
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    ABSTRACT: For each pair of algebraic numbers (x,y) and each fieldF, the complexity of computing the Tutte polynomialT(M;x,y) of a matroidMrepresentable overFis determined. This computation is found to be#P-complete except when (x−1)(y−1)=1 or when |F| divides (x−1)(y−1) and (x,y) is one of the seven points (0,−1), (−1,0), (i,−i), (−i,i), (j,j2), (j2,j) or (−1,−1), wherej=e2πi/3. Expressions are given for the Tutte polynomial in the exceptional cases. These expressions involve the bicycle dimension ofMoverF. A related result determines when this bicycle dimension is well defined.
    J. Comb. Theory, Ser. B. 01/1998; 74:378-396.
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    Neal Brand, Margaret Morton, Dirk Vertigan
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    ABSTRACT: The growth rate of infinite planar 1--ended graphs is considered. For those graphs which are also concentric a recurrence relation is given which determines the growth rate. In the more general case lower bounds on the growth rate are given. In both the concentric and the general cases, the formulae involve the local condition of excess at a vertex. 1 Introduction Graphs considered in this paper are infinite, planar and of bounded degree unless otherwise indicated. The symbols V (Gamma) and E (Gamma) will denote respectively the vertex set and the edge set. We do not in general restrict to simple graphs, but allow multiedges and loops. A loop contributes two to the degree of the incident vertex while each multiedge contributes one to the degree of each incident vertex. An infinite graph is 1--ended if the removal of a finite set of vertices yields at most one infinite component. Knowing the growth rate of the different classes of infinite planar graphs is a key concept in a possible...
    09/1997;

Publication Stats

559 Citations
12.67 Total Impact Points

Institutions

  • 1970–2009
    • Louisiana State University
      • Department of Mathematics
      Baton Rouge, Louisiana, United States
  • 2003
    • Tel Aviv University
      Tell Afif, Tel Aviv, Israel
  • 1995–2000
    • University of Waterloo
      • Department of Combinatorics & Optimization
      Waterloo, Ontario, Canada
  • 1990–1995
    • University of Oxford
      • Mathematical Institute
      Oxford, England, United Kingdom