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  • 1. Aaghabali, M.
    et al.
    Akbari, S.
    Friedland, S.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tajfirouz, Z.
    Upper bounds on the number of perfect matchings and directed 2-factors in graphs with given number of vertices and edges2015In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 45, p. 132-144Article in journal (Refereed)
    Abstract [en]

    We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on 2n vertices. The upper bound is sharp for even n. For odd n we state a conjecture on a sharp upper bound.

  • 2. Aas, Erik
    et al.
    Ayyer, Arvind
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Limiting directions for random walks in classical affine Weyl groupsManuscript (preprint) (Other academic)
    Abstract [en]

    Let be a finite Weyl group and the corresponding affine Weyl group. A random element of can be obtained as a reduced random walk on the alcoves of . By a theorem of Lam (Ann. Probab. 2015), such a walk almost surely approaches one of many directions. We compute these directions when is , and and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the -TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.

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  • 3. Adamaszek, Michal
    et al.
    Barmak, Jonathan Ariel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On a lower bound for the connectivity of the independence complex of a graph2011In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 21, p. 2566-2569Article in journal (Refereed)
    Abstract [en]

    Aharoni, Berger and Ziv proposed a function which is a lower bound for the connectivity of the independence complex of a graph. They conjectured that this bound is optimal for every graph. We give two different arguments which show that the conjecture is false.

  • 4.
    Adler, Mark
    et al.
    Brandeis Univ, Dept Math, Waltham, MA 02254 USA..
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    van Moerbeke, Pierre
    UCLouvain, Dept Math, Louvain, Belgium.;Brandeis Univ, Waltham, MA 02254 USA..
    A SINGULAR TOEPLITZ DETERMINANT AND THE DISCRETE TACNODE KERNEL FOR SKEW-AZTEC RECTANGLES2022In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 32, no 2, p. 1234-1294Article in journal (Refereed)
    Abstract [en]

    Random tilings of geometrical shapes with dominos or lozenges have been a rich source of universal statistical distributions. This paper deals with domino tilings of checker board rectangular shapes such that the top two and bottom two adjacent squares have the same orientation and the two most left and two most right ones as well. It forces these so-called "skew-Aztec rectangles" to have cuts on either side. For large sizes of the domain and upon an appropriate scaling of the location of the cuts, one finds split tacnodes between liquid regions with two distinct adjacent frozen phases descending into the tacnode. Zooming about such split tacnodes, filaments appear between the liquid patches evolving in a bricklike sea of dimers of another type. This work shows that the random fluctuations in a neighborhood of the split tacnode are governed asymptotically by the discrete tacnode kernel, providing strong evidence that this kernel is a universal discrete-continuous limiting kernel occurring naturally whenever we have doubly interlacing patterns. The analysis involves the inversion of a singular Toeplitz matrix which leads to considerable difficulties.

  • 5.
    Agdur, Vilhelm
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    A classification of well-behaved graph clustering schemesManuscript (preprint) (Other academic)
  • 6.
    Agdur, Vilhelm
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Community detection: Lower bounds and axiomatic classification2023Licentiate thesis, comprehensive summary (Other academic)
  • 7.
    Agdur, Vilhelm
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Skerman, Fiona
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Kamčev, Nina
    University of Zagreb.
    Universal lower bound for community structure of sparse graphsManuscript (preprint) (Other academic)
  • 8.
    Ahmad, Naseer
    Blekinge Institute of Technology, School of Engineering, Department of Telecommunication Systems.
    Security Issues in Wireless Systems2009Independent thesis Advanced level (degree of Master (One Year))Student thesis
    Abstract [en]

    ireless Communication is one of the fields of Telecommunications which is growing with the tremendous speed. With the passage of time wireless communication devices are becoming more and more common. It is not only the technology of business but now people are using it to perform their daily tasks, be it for calling, shopping, checking their emails or transfer their money. Wireless communication devices include cellular phones, cordless phones and satellite phones, smart phones like Personal Digital Assistants (PDA), two way pagers, and lots of their devices are on their way to improve this wireless world. In order to establish two way communications, a wireless link may be using radio waves or Infrared light. The Wireless communication technologies have become increasingly popular in our everyday life. The hand held devices like Personal Digital Assistants (PDA) allow the users to access calendars, mails, addresses, phone number lists and the internet. Personal digital assistants (PDA) and smart phones can store large amounts of data and connect to a broad spectrum of networks, making them as important and sensitive computing platforms as laptop PCs when it comes to an organization’s security plan. Today’s mobile devices offer many benefits to enterprises. Mobile phones, hand held computers and other wireless systems are becoming a tempting target for virus writers. Mobile devices are the new frontier for viruses, spam and other potential security threats. Most viruses, Trojans and worms have already been created that exploit vulnerabilities. With an increasing amount of information being sent through wireless channels, new threats are opening up. Viruses have been growing fast as handsets increasingly resemble small computers that connect with each other and the internet. Hackers have also discovered that many corporate wireless local area networks (WLAN) in major cities were not properly secured. Mobile phone operators say that it is only a matter of time before the wireless world is hit by the same sorts of viruses and worms that attack computer software.

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  • 9.
    Aichholzer, Oswin
    et al.
    Graz University of Technology, Austria.
    Anna, Brötzner
    Malmö University, Faculty of Technology and Society (TS), Department of Computer Science and Media Technology (DVMT).
    Bicolored Order Types2024In: Computing in Geometry and Topology, ISSN 2750-7823, Vol. 3, no 2Article in journal (Refereed)
    Abstract [en]

    In their seminal work on Multidimensional Sorting, Goodman and Pollack introduced the so-called order type, which for each ordered triple of a point set in the plane gives its orientation, clockwise or counterclockwise. This information is sufficient to solve many problems from discrete geometry where properties of point sets do not depend on the exact coordinates of the points but only on their relative positions. Goodman and Pollack showed that an efficient way to store an order type in a matrix λ of quadratic size (w.r.t. the number of points) is to count for every oriented line spanned by two points of the set how many of the remaining points lie to the left of this line. We generalize the concept of order types to bicolored point sets (every point has one of two colors). The bicolored order type contains the orientation of each bicolored triple of points, while no information is stored for monochromatic triples. Similar to the uncolored case, we store the number of blue points that are to the left of an oriented line spanned by two red points or by one red and one blue point in λB. Analogously the number of red points is stored in λR. As a main result, we show that the equivalence of the information contained in the orientation of all bicolored point triples and the two matrices λB and λR also holds in the colored case. This is remarkable, as in general the bicolored order type does not even contain sufficient information to determine all extreme points (points on the boundary of the convex hull of the point set).We then show that the information of a bicolored order type is sufficient to determine whether the two color classes can be linearly separated and how one color class can be sorted around a point of the other color class. Moreover, knowing the bicolored order type of a point set suffices to find bicolored plane perfect matchings or to compute the number of crossings of the complete bipartite graph drawn on a bicolored point set in quadratic time.

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  • 10. Akbari, Saieed
    et al.
    Friedland, Shmuel
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zare, Sanaz
    On 1-sum flows in undirected graphs2016In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 31, p. 646-665Article in journal (Refereed)
    Abstract [en]

    Let G = (V, E) be a simple undirected graph. For a given set L subset of R, a function omega: E -> L is called an L-flow. Given a vector gamma is an element of R-V , omega is a gamma-L-flow if for each v is an element of V, the sum of the values on the edges incident to v is gamma(v). If gamma(v) = c, for all v is an element of V, then the gamma-L-flow is called a c-sum L-flow. In this paper, the existence of gamma-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L* := L \ {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum R* -flow or a 1-sum Z* -flow are characterized. It is also shown that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.

  • 11.
    Alanwar, Amr
    et al.
    School of Computation, Information and Technology, Technical University of Munich, School of Computation, Information and Technology, Technical University of Munich; School of Computer Science and Engineering, Constructor University, School of Computer Science and Engineering, Constructor University.
    Jiang, Frank
    KTH, School of Electrical Engineering and Computer Science (EECS), Intelligent systems, Decision and Control Systems (Automatic Control).
    Amin, Samy
    School of Computer Science and Engineering, Constructor University, School of Computer Science and Engineering, Constructor University.
    Johansson, Karl H.
    KTH, School of Electrical Engineering and Computer Science (EECS), Intelligent systems, Decision and Control Systems (Automatic Control).
    Logical Zonotopes: A Set Representation for the Formal Verification of Boolean Functions2023In: 2023 62nd IEEE Conference on Decision and Control, CDC 2023, Institute of Electrical and Electronics Engineers (IEEE) , 2023, p. 60-66Conference paper (Refereed)
    Abstract [en]

    A logical zonotope, which is a new set representation for binary vectors, is introduced in this paper. A logical zonotope is constructed by XORing a binary vector with a combination of other binary vectors called generators. Such a zonotope can represent up to 2γ binary vectors using only γ generators. It is shown that logical operations over sets of binary vectors can be performed on the zonotopes' generators and, thus, significantly reduce the computational complexity of various logical operations (e.g., XOR, NAND, AND, OR, and semi-tensor products). Similar to traditional zonotopes' role in the formal verification of dynamical systems over real vector spaces, logical zonotopes can efficiently analyze discrete dynamical systems defined over binary vector spaces. We illustrate the approach and its ability to reduce the computational complexity in two use cases: (1) encryption key discovery of a linear feedback shift register and (2) safety verification of a road traffic intersection protocol.

  • 12.
    Alexandersson, Per
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Getachew Kebede, Frether
    An involution on derangements preserving excedances and right-to-left minima2023In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 86, no 3, p. 387-413Article in journal (Refereed)
    Abstract [en]

    We give a bijective proof of a result by Mantaci and Rakotondrajao from 2003, regarding even and odd derangements with a fixed number of excedances. We refine their result by also considering the set of right-to-left minima

  • 13.
    Alexandersson, Per
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kebede, Frether Getachew
    Fufa, Samuel Asefa
    Qiu, Dun
    Pattern-Avoidance and Fuss-Catalan Numbers2023In: Journal of Integer Sequences, E-ISSN 1530-7638, Vol. 26, article id 23.4.2Article in journal (Refereed)
    Abstract [en]

    We study a subset of permutations where entries are restricted to having the same remainder as the index, modulo some integer k ≥ 2. We show that by also imposing the classical 132- or 213-avoidance restriction on the permutations, we recover the Fuss–Catalan numbers and some special cases of the Raney numbers. Surprisingly, an analogous statement also holds when we impose the mod k restriction on a Catalan family of subexcedant functions. Finally, we completely enumerate all combinations of mod-k-alternating permutations that avoid two patterns of length 3. This is analogous to the systematic study by Simion and Schmidt, of permutations avoiding two patterns of length 3. 

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  • 14.
    Alexandersson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Uhlin, Joakim
    Refined Catalan and Narayana cyclic sievingManuscript (preprint) (Other academic)
    Abstract [en]

    We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of “ears”, non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.

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  • 15.
    Alexandersson, Per
    et al.
    Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden.
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Uhlin, Joakim
    Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden.
    Refined Catalan and Narayana cyclic sieving2021In: Combinatorial Theory, E-ISSN 2766-1334, Vol. 1, no 0Article in journal (Refereed)
    Abstract [en]

    We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B . Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.

  • 16.
    Alexandersson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Panova, Greta
    LLT polynomials, chromatic quasisymmetric functions and graphs with cycles2018In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 341, no 12, p. 3453-3482Article in journal (Refereed)
    Abstract [en]

    We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their e-positivity. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the noncircular combinatorics, we prove several statements regarding the e-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the e-coefficients for the line graph and the cycle graph for both families. We believe that certain e-positivity conjectures hold in all these families above. Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall-Littlewood polynomials. 

  • 17.
    Alexandersson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sulzgruber, Robin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    P-partitions and p-positivity2019In: FPSAC 2019 - 31st International Conference on Formal Power Series and Algebraic Combinatorics, Formal Power Series and Algebraic Combinatorics , 2019Conference paper (Refereed)
    Abstract [en]

    Using the combinatorics of a-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse P-partitions expand positively into quasisymmetric power sums. Consequently any nonnegative linear combination of such functions is p-positive whenever it is symmetric. We apply this method to derive positivity results for chromatic quasisymmetric functions and unicellular LLT polynomials. 

  • 18.
    Alsaody, Seidon
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Chernousov, Vladimir
    Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada..
    Pianzola, Arturo
    Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada.;Ctr Altos Estudios Ciencias Exactas, Ave Mayo 866, RA-1084 Buenos Aires, DF, Argentina..
    On the Tits-Weiss conjecture and the Kneser-Tits conjecture for E-7,1(78) and E-8,2(78) (With an Appendix by R. M. Weiss)2021In: Forum of Mathematics, Sigma, ISSN 2050-5094, Vol. 9, article id e75Article in journal (Refereed)
    Abstract [en]

    We prove that the structure group of any Albert algebra over an arbitrary field is R-trivial. This implies the TitsWeiss conjecture for Albert algebras and the Kneser-Tits conjecture for isotropic groups of type E-7,1(78), E-8,2(78). As a further corollary, we show that some standard conjectures on the groups of R-equivalence classes in algebraic groups and the norm principle are true for strongly inner forms of type E-1(6).

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  • 19.
    Amini, Nima
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Combinatorics and zeros of multivariate polynomials2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of five papers in algebraic and enumerative combinatorics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special evaluations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a central conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur\'an inequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs. In Paper C we prove that the multivariate $d$-matching polynomial is hyperbolic for arbitrary multigraphs, in particular answering a question by Hall, Puder and Sawin. We also provide a hypergraphic generalization of a classical theorem by Heilmann and Lieb regarding the real-rootedness of the matching polynomial of a graph. In Paper D we establish a number of equidistributions between Mahonian statistics which are given by conic combinations of vincular pattern functions of length at most three, over permutations avoiding a single classical pattern of length three. In Paper E we find necessary and sufficient conditions for a candidate polynomial to be complemented to a cyclic sieving phenomenon (without regards to combinatorial context). We further take a geometric perspective on the phenomenon by associating a convex rational polyhedral cone which has integer lattice points in correspondence with cyclic sieving phenomena. We find the half-space description of this cone and investigate its properties.

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  • 20.
    Amini, Nima
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Spectrahedrality of hyperbolicity cones of multivariate matching polynomials2018In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Article in journal (Refereed)
    Abstract [en]

    The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

  • 21.
    Amini, Nima
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Stable multivariate generalizations of matching polynomialsManuscript (preprint) (Other academic)
    Abstract [en]

    The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan d-coverings, Hall, Puder and Sawin introduced the d-matching polynomial of a graph G, defined as the uniform average of matching polynomials over the set of d-sheeted covering graphs of G. We prove that a natural multivariate version of the d-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the d-matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem.

  • 22.
    Amini, Nima
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Alexandersson, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The cone of cyclic sieving phenomena2019In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 342, no 6, p. 1581-1601Article in journal (Refereed)
    Abstract [en]

    We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits.

    Reiner et.al. showed that every cyclic action gives rise to a unique polynomial (mod q^n-1) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic gives rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.

  • 23.
    Anastasiou, Aspasia
    et al.
    KTH, Centres, Nordic Institute for Theoretical Physics NORDITA.
    Borsten, L.
    Dublin Inst Adv Studies, Sch Theoret Phys, 10 Burlington Rd, Dublin 4, Ireland..
    Duff, M. J.
    Imperial Coll London, Blackett Lab, Theoret Phys, London SW7 2AZ, England.;Univ Oxford, Math Inst, Radcliffe Observ Quarter, Andrew Wiles Bldg,Woodstock Rd, Oxford OX2 6GG, England..
    Marrani, A.
    Museo Stor Fis, Via Panisperna 89A, I-00184 Rome, Italy.;Ctr Studi & Ric Enrico Fermi, Via Panisperna 89A, I-00184 Rome, Italy.;Univ Padua, Dipartimento Fis & Astron Galileo Galilei, Via Marzolo 8, I-35131 Padua, Italy.;INFN, Sez Padova, Via Marzolo 8, I-35131 Padua, Italy..
    Nagy, S.
    Ctr Astron & Particle Theory, Univ Pk, Nottingham NG7 2RD, England..
    Zoccali, M.
    Imperial Coll London, Blackett Lab, Theoret Phys, London SW7 2AZ, England..
    The mile high magic pyramid2019In: NONASSOCIATIVE MATHEMATICS AND ITS APPLICATIONS / [ed] Vojtechovsky, P Bremner, MR Carter, JS Evans, AB Huerta, J Kinyon, MK Moorhouse, GE Smith, JDH, AMER MATHEMATICAL SOC , 2019, p. 1-27Conference paper (Refereed)
    Abstract [en]

    Using a unified formulation of N = 1, 2, 4, 8, super Yang-Mills theories in D = 3 spacetime dimensions with fields valued respectively in R, C, H, O, it was shown that tensoring left and right multiplets yields a Freudenthal magic square of D = 3 supergravities. When tied in with the more familiar R, C, H, O description of super Yang-Mills in D = 3, 4, 6, 10 this results in a magic pyramid of supergravities: the known 4x4 magic square at the base in D = 3, a 3x3 square in D = 4, a 2x2 square in D = 6 and Type II supergravity at the apex in D = 10.

  • 24.
    Andersson, Håkan
    Linköping University, Department of Electrical Engineering. Linköping University, The Institute of Technology.
    Error-Correcting Codes Based on Chaotic Dynamical Systems1998Doctoral thesis, monograph (Other academic)
    Abstract [en]

    This dissertation treats a novel class of error-correcting codes based on chaotic dynamical systems. The codes are defined over a continuous alphabet whereas the information that is to be transmitted belongs to a discrete set of symbols. Simple expressions can be given for the encoders, and the codewords can be described by a parity-check relation. However, the most interesting approach is to view the codewords as orbits of iterated dynamical systems described by integer matrices.

    Under some rather natural assumptions, the codes are shown to be group codes. The minimum distance is proved to be well-defined and strictly greater than zero. An algorithm for calculating it is also given. Initially, no robust sliding-window encoder inverses exist, but this deficiency is remedied by the introduction of fractal signal sets. The problem of catastrophic encoders is also solved by the introduction of these totally disconnected signal sets.

    Decoding strategies are discussed, and it is shown why the Viterbi algorithm does not work in higher dimensions for this type of codes. So-called list decoding emerges as a good alternative and its merits are considered. Simulations and comparisons with binary antipodal signaling are performed. The setting of the work is in two dimensions. However, the strength of this code construction is that it easily generalizes to higher dimensions.

  • 25.
    Andren, Daniel
    et al.
    Umeå University.
    Hellström, Lars
    Umeå University.
    Markström, Klas
    Umeå University.
    Fast multiplication of matrices over a finitely generated semiring2008In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, no 6, p. 230-234Article in journal (Refereed)
  • 26.
    Andren, Lina J.
    et al.
    Malardalen Univ, Sweden; Mittag Leffler Inst, Sweden.
    Casselgren, Carl Johan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Mittag Leffler Inst, Sweden.
    Markstrom, Klas
    Umea Univ, Sweden; Mittag Leffler Inst, Sweden.
    Restricted completion of sparse partial Latin squares2019In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 28, no 5, p. 675-695Article in journal (Refereed)
    Abstract [en]

    An n x n Martial Latin square P is called dense if each row and column has at mostan non-empty cells and each symboloccurso n times in P. An x n arrayA where each cell contains subset of {1, ..., n} is a (beta n, beta n, beta n)-array if each symbol occurs at most beta n times in each row and column and each cell contains a set of size at most beta n. Combining the notions of completing partial Latin squared and avoiding arrays, we prose that there are constants alpha, beta amp;gt; 0 such that, for every positive integer n, if P is an alpha-dense n x n partial a square, A is an n x n (beta n, beta n, beta n)-array and no cell of P contains a symbol that ppears in the corresponing cell of A, then there is a completiong of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty of P, and for each i, j satisfying 1 amp;lt;= i, j, amp;lt;= n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

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  • 27. Andren, Lina J.
    et al.
    Casselgren, Carl Johan
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Restricted completion of sparse partial Latin squares2019In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 28, no 5, p. 675-695Article in journal (Refereed)
    Abstract [en]

    An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

  • 28.
    Andren, Lina J.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Casselgren, Carl Johan
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoiding Arrays of Odd Order by Latin Squares2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, p. 184-212Article in journal (Refereed)
    Abstract [en]

    We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m <= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -> infinity.

  • 29.
    Andriantiana, Eric O. D.
    et al.
    Rhodes Univ, Dept Math Pure & Appl, POB 94, ZA-6140 Grahamstown, South Africa..
    Misanantenaina, Valisoa Razanajatovo
    Stellenbosch Univ, Dept Math Sci, Private Bag X1, ZA-7602 Matieland, South Africa..
    Wagner, Stephan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Stellenbosch Univ, Dept Math Sci, Private Bag X1, ZA-7602 Matieland, South Africa.
    Extremal trees with fixed degree sequence2021In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 28, no 1, article id P1.1Article in journal (Refereed)
    Abstract [en]

    The greedy tree G(D) and the M-tree M(D) are known to be extremal among trees with degree sequence D with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which G(D) or M(D) is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence D to the set of trees whose degree sequence is majorised by a given sequence D, which also has a number of applications.

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  • 30.
    Andriantiana, Eric Ould Dadah
    et al.
    Rhodes Univ, Dept Math Pure & Appl, POB 94, ZA-6140 Grahamstown, South Africa..
    Wagner, Stephan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Probability Theory and Combinatorics. Stellenbosch Univ, Dept Math Sci, Private Bag X1, ZA-7602 Matieland, South Africa..
    Trees with minimum number of infima closed sets2022In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 345, no 5, article id 112793Article in journal (Refereed)
    Abstract [en]

    Let T be a rooted tree, and V(T) its set of vertices. A subset X of V(T) is called an infima closed set of T if for any two vertices u, v is an element of X, the first common ancestor of u and v is also in X. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with n vertices is also provided.

  • 31.
    Andrén, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the Ising problem and some matrix operations2007Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour.

    Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices).

    In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice.

    To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy.

    The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function.

    The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations.

    We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.

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  • 32.
    Andrén, Daniel
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hellström, Lars
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the complexity of matrix reduction over finite fields2007In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 39, no 4, p. 428-452Article in journal (Refereed)
    Abstract [en]

    We study matrix elimination over finite fields, and present an algorithm which is asymptotically faster than the traditional Gauss--Jordan elimination. We also bound the average and worst-case complexity for the problem, proving that our algorithm is close to being optimal, and show related concentration results for random matrices.

    Next we present the results of a large computational study of the complexities for small matrices and fields. Here we determine the exact distribution of the complexity for matrices from $\mathrm{GL}_{n}(\mathbb{F}_{q})$, with $n$ an $q$ small

    Finally we consider an extension of the problems studied for finite fields to finite semifields. We give a conjecture on the behaviour of a natural analogue of $\mathrm{GL}_{n}$ for semifields and prove this for a certain class of semifields.

  • 33.
    Andrén, Lina J.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoidability by Latin squares of arrays of even orderManuscript (preprint) (Other academic)
    Abstract [en]

    We prove that for any k and any 2k × 2k array A such that no cell in A contains more than   k/2550 symbols, and no symbol occurs more than k/2550 times in any row or column, there is a Latin square such that no 2550cell in the Latin square contains a symbol that occurs in the corresponding cell in A. This proves a conjecture of Häggkvist [8] in the special case of arrays with even side.

  • 34.
    Andrén, Lina J.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoidability of random arraysManuscript (preprint) (Other academic)
    Abstract [en]

    An n×n array that in each cell contains a subset of the symbols 1, . . . , n is avoidable if there exists a Latin square of order n such that no cell in the Latin square contains a symbol which belongs to the set of symbols in the corresponding cell of the array. Some results on deterministic conditions for avoidability of arrays have been found, but here we study the problem of having an array with randomly assigned subsets of C in its cells. This is equivalent to the problem of list-edge-coloring  with randomly assigned lists from the set {1, . . . , n}. We show that an array where each symbol appears in each cell with probability p will be avoidable with very high probability even if p is such that the expected number of symbols forbidden in each cell is slightly higher than what deterministic theorems can prove is avoidable.

  • 35.
    Andrén, Lina J.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoiding (m, m, m)-arrays of order n = 2kManuscript (preprint) (Other academic)
    Abstract [en]

    An (m, m, m)-array of order n is an n × n array such that each cell is assigned a set of at most m symbols from {1,...,n} such that no symbol occurs more than m times in any row or column. An (m,m,m)- array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant γ such that if m ≤ γ2k, then any (m,m,m)-array of order 2k is avoidable. Such a constant γ has been conjectured to exist for all n by Häggkvist.

  • 36.
    Andrén, Lina J.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On Latin squares and avoidable arrays2010Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of the four papers listed below and a survey of the research area.

    I Lina J. Andrén: Avoiding (m, m, m)-arrays of order n = 2k

    II Lina J. Andrén: Avoidability of random arrays

    III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order

    IV Lina J. Andrén, Carl Johan Casselgren and Lars-Daniel Öhman: Avoiding arrays of odd order by Latin squares

    Papers I, III and IV are all concerned with a conjecture by Häggkvist saying that there is a constant c such that for any positive integer n, if m ≤ cn, then for every n × n array A of subsets of {1, . . . , n} such that no cell contains a set of size greater than m, and none of the elements 1, . . . , n belongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1, . . . , n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A. Such a Latin square is said to avoid A. In Paper I, the conjecture is proved in the special case of order n = 2k . Paper III improves on the techniques of Paper I, expanding the proof to cover all arrays of even order. Finally, in Paper IV, similar methods are used together with a recoloring theorem to prove the conjecture for all orders. Paper II considers another aspect of the problem by asking to what extent way a deterministic result concerning the existence of Latin squares that avoid certain arrays can be used when the sets in the array are assigned randomly.

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  • 37.
    Andrén, Lina J.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoiding arrays of odd order by Latin squaresManuscript (preprint) (Other academic)
    Abstract [en]

    We prove that there exists a constant c such that for each pos- itive integer k every (2k+1)×(2k+1) array A on the symbols 1,...,2k+1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1)×(2k+1) Latin square S on the symbols 1,...,2k+1 such that for each cell (i, j) in S the symbol in (i, j) does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist.

  • 38. Ardila, Federico
    et al.
    Supina, Mariel
    Vindas-Meléndez, Andrés R.
    The equivariant Ehrhart theory of the permutahedron2020In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 148, no 12, p. 5091-5107Article in journal (Refereed)
    Abstract [en]

    Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case.

  • 39.
    Arkin, Esther M
    et al.
    Department of Applied Mathematics and Statistics, Stony Brook University, USA .
    Dieckmann, Claudia
    Institute of Computer Science, Freie Universität Berlin, Germany .
    Knauer, Christian
    Institute of Computer Science, Universität Bayreuth, Germany .
    Mitchell, Joseph SB
    Department of Applied Mathematics and Statistics, Stony Brook University, USA .
    Polishchuk, Valentin
    Helsinki Institute for Information Technology, CS Dept, University of Helsinki, Finland .
    Schlipf, Lena
    Institute of Computer Science, Freie Universität Berlin, Germany .
    Yang, Shang
    Department of Computer Science, Stony Brook University, USA .
    Convex transversals2014In: Computational Geometry, ISSN 0925-7721, Vol. 47, no 2, p. 224-239Article in journal (Refereed)
    Abstract [en]

    We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): “Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?”

    We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are regular polygons. We also show that in 3D the stabbing problem is hard when the sets are balls. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments in 2D if the vertices of the transversal are restricted to a given set of points. Our algorithm also finds a convex stabber of the maximum number of a set of convex pseudodisks in the plane.

    The stabbing problem is related to “convexity” of point sets measured as the minimum distance by which the points must be shifted in order to arrive in convex position; we give a PTAS to find the minimum shift in 2D, and a 2-approximation in any dimension. We also consider stabbing with vertices of a regular polygon – a problem closely related to approximate symmetry detection.

  • 40. Aryapoor, Masood
    et al.
    Mahmoodian, Ebadollah S.
    On uniformly generating Latin squares2011In: Bulletin of the Institute of Combinatorics and its Applications, ISSN 1183-1278, Vol. 62, p. 48-58Article in journal (Refereed)
  • 41.
    Asratian, Armen
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Diskret matematik2020 (ed. 1)Book (Other academic)
    Abstract [sv]

    Den här boken är främst avsedd för grundläggande kurser i diskret matematik vid universitet och högskolor. Framför allt riktar den sig till första- och andraårsstudenter på data-, matematik-, civilingenjörs- och högskoleingenjörsprogrammen.

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  • 42.
    Asratian, Armen
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Casselgren, Carl Johan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. University of Southern Denmark, Denmark.
    Solution of Vizings Problem on Interchanges for the case of Graphs with Maximum Degree 4 and Related Results2016In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 82, no 4, p. 350-373Article in journal (Refereed)
    Abstract [en]

    Let G be a Class 1 graph with maximum degree 4 and let t amp;gt;= 5 be an integer. We show that any proper t-edge coloring of G can be transformed to any proper 4-edge coloring of G using only transformations on 2-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper 5-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent. (C) 2015 Wiley Periodicals, Inc.

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  • 43.
    Asratian, Armen
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Casselgren, Carl Johan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Petrosyan, Petros A.
    Yerevan State Univ, Armenia; Natl Acad Sci, Armenia.
    Cyclic deficiency of graphs2019In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 266, p. 171-185Article in journal (Refereed)
    Abstract [en]

    A proper edge coloring of a graph G with colors 1, 2, . . . , t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. In this paper we introduce and investigate a new notion, the cyclic deficiency of a graph G, defined as the minimum number of pendant edges whose attachment to G yields a graph admitting a cyclic interval coloring; this number can be considered as a measure of closeness of G of being cyclically interval colorable. We determine or bound the cyclic deficiency of several families of graphs. In particular, we present examples of graphs of bounded maximum degree with arbitrarily large cyclic deficiency, and graphs whose cyclic deficiency approaches the number of vertices. Finally, we conjecture that the cyclic deficiency of any graph does not exceed the number of vertices, and we present several results supporting this conjecture. (C) 2018 Elsevier B.V. All rights reserved.

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  • 44.
    Asratian, Armen
    et al.
    Linköping University, Department of Mathematics, Algebra, Geometry and Discrete Mathematics. Linköping University, Faculty of Science & Engineering.
    Casselgren, Carl Johan
    Linköping University, Department of Mathematics, Algebra, Geometry and Discrete Mathematics. Linköping University, Faculty of Science & Engineering.
    Petrosyan, Petros A.
    Yerevan State Univ, Armenia.
    Decomposing graphs into interval colorable subgraphs and no-wait multi-stage schedules2023In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 335, p. 25-35Article in journal (Refereed)
    Abstract [en]

    A graph G is called interval colorable if it has a proper edge coloring with colors 1, 2, 3, ... such that the colors of the edges incident to every vertex of G form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph G, denoted theta int(G), which is the minimum number of interval colorable edge-disjoint subgraphs of G whose union is G. Our investigation is motivated by scheduling problems with compactness require-ments, in particular, problems whose solution may consist of several schedules, but where each schedule must not contain any waiting periods or idle times for all involved parties. We first prove that every connected properly 3-edge colorable graph with maximum degree 3 is interval colorable, and using this result, we deduce an upper bound on theta int(G) for general graphs G. We demonstrate that this upper bound can be improved in the case when G is bipartite, planar or complete multipartite and consider some applications in timetabling.

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  • 45.
    Asratian, Armen
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Casselgren, Carl Johan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Petrosyan, Petros A.
    Yerevan State University, Armenia; National Academic Science, Armenia.
    Some results on cyclic interval edge colorings of graphs2018In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 87, no 2, p. 239-252Article in journal (Refereed)
    Abstract [en]

    A proper edge coloring of a graph G with colors 1,2,,t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree (G)4 admits a cyclic interval (G)-coloring if for every vertex v the degree dG(v) satisfies either dG(v)(G)-2 or dG(v)2. We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for (a,b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)-biregular graphs as well as all (2r-2,2r)-biregular (r2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.

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  • 46.
    Asratian, Armen
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Granholm, Jonas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Khachatryan, Nikolay K.
    Synopsys Armenia CJSC, Armenia.
    A localization method in Hamiltonian graph theory2021In: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 148, p. 209-238Article in journal (Refereed)
    Abstract [en]

    The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to Dirac (1952), Ore (1960), Jung (1978), and Nash-Williams (1971). The idea was to show that the global concept of Hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. (The ball of radius r centered at a vertex u is a subgraph of G induced by the set of vertices whose distances from u do not exceed r.) Such results are called localization theorems and present a possibility to extend known classes of finite Hamiltonian graphs. In this paper we formulate a general approach for finding localization theorems and use this approach to formulate local analogues of well-known results of Bauer et al. (1989), Bondy (1980), Haggkvist and Nicoghossian (1981), and Moon and Moser (1963). Finally we extend two of our results to infinite locally finite graphs and show that they guarantee the existence of Hamiltonian curves, introduced by Kundgen, Li and Thomassen (2017). (c) 2020 Elsevier Inc. All rights reserved.

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  • 47.
    Asratian, Armen
    et al.
    Linköping University, Department of Mathematics, Algebra, Geometry and Discrete Mathematics. Linköping University, Faculty of Science & Engineering.
    Granholm, Jonas
    Linköping University, Department of Mathematics, Algebra, Geometry and Discrete Mathematics. Linköping University, Faculty of Science & Engineering.
    Khachatryan, Nikolay K.
    Synopsys Armenia CJSC, Armenia.
    Some local–global phenomena in locally finite graphs2021In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 293, p. 166-176Article in journal (Refereed)
    Abstract [en]

    In this paper we present some results for a connected infinite graph G with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of G. (For a vertex w of a graph G the ball of radius r centered at w is the subgraph of G induced by the set Mr(w) of vertices whose distance from w does not exceed r). In particular, we prove that if every ball of radius 2 in G is 2-connected and G satisfies the condition dG(u)+dG(v)≥|M2(w)|−1 for each path uwv in G, where u and v are non-adjacent vertices, then G has a Hamiltonian curve, introduced by Kündgen et al. (2017). Furthermore, we prove that if every ball of radius 1 in G satisfies Ore’s condition (1960) then all balls of any radius in G are Hamiltonian.

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  • 48.
    Asratian, Armen S.
    et al.
    Linköping University, Linköping, Sweden.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vandenbussche, Jennifer
    Southern Polytechnic State University, Marietta, Georgia.
    West, Douglas B.
    University of Illinois, Urbana, Illinois.
    Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs2009In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 61, no 2, p. 88-97Article in journal (Refereed)
    Abstract [en]

    An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a subgraph.

  • 49. Atkins, Ross
    et al.
    Rombach, Puck
    Skerman, Fiona
    Heilbronn Institute University of Bristol Bristol, U.K..
    Guessing Numbers of Odd Cycles2017In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 24, no 1, article id P1.45Article in journal (Refereed)
    Abstract [en]

    For a given number of colours, s, the guessing number of a graph is the base s logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the n-vertex cycle graph Cn is n/2. It is known that the guessing number equals n/2 whenever n is even or s is a perfect square. We show that, for any given integer s≥2, if a is the largest factor of s less than or equal to √s, for sufficiently large odd n, the guessing number of Cn with s colours is (n−1)/2+logs(a). This answers a question posed by Christofides and Markström in 2011.We also present an explicit protocol which achieves this bound for every n. Linking this to index coding with side information, we deduce that the information defect of Cn with s colours is (n+1)/2−logs(a) for sufficiently large odd n. 

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  • 50.
    Atserias, Albert
    et al.
    Univ Politecn Cataluna, Dept Comp Sci, Barcelona, Spain..
    Bonacina, Ilario
    Univ Politecn Cataluna, Dept Comp Sci, Barcelona, Spain..
    de Rezende, Susanna F.
    KTH, School of Electrical Engineering and Computer Science (EECS).
    Lauria, Massimo
    Sapienza Univ Roma, Dept Stat Sci, Rome, Italy..
    Nordström, Jakob
    KTH, School of Electrical Engineering and Computer Science (EECS).
    Razborov, Alexander
    Univ Chicago, Chicago, IL 60637 USA.;Steklov Math Inst, Moscow, Russia..
    Clique Is Hard on Average for Regular Resolution2018In: STOC'18: PROCEEDINGS OF THE 50TH ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING / [ed] Diakonikolas, I Kempe, D Henzinger, M, ASSOC COMPUTING MACHINERY , 2018, p. 866-877Conference paper (Refereed)
    Abstract [en]

    We prove that for k << (4)root n regular resolution requires length n(Omega(k)) to establish that an Erdos Renyi graph with appropriately chosen edge density does not contain a k-clique. This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional n(Omega(k)) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

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