Publications (35)24.11 Total impact
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ABSTRACT: It is found that every Dowling geometry over a finite group G is also representable over some ring (or equivalently, over some skew partial field) if and only if G is a Frobenius Complement. These groups have been completely characterized.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that, for every positive integer k, there is an integer N such that every 4connected nonplanar 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 3cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3connected crossingcritical 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  [Show abstract] [Hide abstract]
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 4connected 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 kpartition triple fan with a complete graph on the vertices of degree k, the kspoke double wheel, the kspoke 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 3connected graphs.  [Show abstract] [Hide abstract]
ABSTRACT: We consider what the implications would be if there were a discrete fundamental model of physics based on locallyfinite selfinteracting 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 selfinteracting information I, called a SpaceCell. Copies of SpaceCell I must be distributed throughout space, at a roughly constant and nearPlanck density, and copies must be created or destroyed as space expands or contracts. We then argue that each SpaceCell is a selfreplicator that can duplicate in times ranging from as fast as nearPlancktimes to as slow as CosmologicalConstanttime which is 10^{61} Plancktimes. 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 lengthC 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 HoleBang Transitions. We then present Cosmological Natural Selection II.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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 coloringo 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 edgewidth at least M, where the edgewidth 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 edgewidth 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 edgewidth, then c(G) 2 (3; 3+")((4; 5). It is also shown that there exist Eulerian triangulations of arbitrarily large edgewidth on nonorientable surfaces whose circular chromatic number is equal to 5.  [Show abstract] [Hide abstract]
ABSTRACT: For each pair of algebraic numbers $(x,y)$, the complexity of computing the Tutte polynomial $T(G;x,y)$ of a planar graph $G$ is determined. This computation is found to be $\overline{\rm\#P}$complete except when $(x1)(y1)=1,2$ or when $(x,y)$ is one of $(1,1)$, $(1,1)$, $(j,j^2)$, or $(j^2,j)$, where $j=e^{2\pi i/3}$, in which case it is polynomial time computable. A corollary gives the computational complexity of various enumeration problems for planar graphs.  [Show abstract] [Hide abstract]
ABSTRACT: It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases. In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these. We define generating functions associated with codes and latroids, and prove identities relating them, generalizing abovementioned generating functions and identities.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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 2coloring of either its vertices or its edges where each color induces a graph of treewidth at most k. Some generalizations are also proved.  [Show abstract] [Hide abstract]
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 treewidth 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.  [Show abstract] [Hide abstract]
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. 
Article: Totally Free Expansions of Matroids.
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ABSTRACT: This paper strengthens the excludedminor characterization of GF(4)representable matroids. In particular, it is shown that there are only finitely many 3connected matroids that are not GF(4)representable and that have no U 2,6 , U 4,6 , P 6 , 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 circuithyperplanes.  [Show abstract] [Hide abstract]
ABSTRACT: In 1971, G. Chartrand, D. Geller and S. Hedetniemi [J. Comb. Theory, Ser. B 10, 1241 (1971; Zbl 0223.05101)] 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 seriesparallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large cliqueminor. Several open questions are raised.  [Show abstract] [Hide abstract]
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 excludedminor result for kregular matroids, where such matroids generalize regular matroids and Whittle's nearregular matroids.  [Show abstract] [Hide abstract]
ABSTRACT: The aim of this paper is to give insight into the behaviour of inequivalent representations of 3connected 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 3connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3connected 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 rankr quaternary and quinternary matroids, the first being the free spike. Finally, Seymour's Splitter Theorem is extended by showing that the sequence of 3connected 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.  [Show abstract] [Hide abstract]
ABSTRACT: The classes of nearregular 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 nearregular and a simple rank{r 6 p 1{matroid and determines all such matroids having these sizes. 1.  [Show abstract] [Hide abstract]
ABSTRACT: LetFbe a field and letNbe a matroid in a class ofFrepresentable matroids that is closed under minors and the taking of duals. ThenNis anFstabilizer for if every representation of a 3connected member of is determined up to elementary row operations and column scaling by a representation of any one of itsNminors. 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 withFstabilizers, 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 rankpreserving weakmap image of any matroid in , then no connected matroid in with anNminor is a strict rankpreserving weakmap image of any 3connected 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, 6minor. Moreover, ifM1andM2are distinct quaternary matroids withU2, 5minors but noU3, 6minors andM1is connected whileM2is 3connected, thenM1is not a rankpreserving weakmap image ofM2. 
Article: Generalized Delta?Y Exchange and
Publication Stats
789  Citations  
24.11  Total Impact Points  
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Institutions

19952015

Louisiana State University
 Department of Mathematics
Baton Rouge, Louisiana, United States


19901995

University of Oxford
 Mathematical Institute
Oxford, England, United Kingdom
