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• 1. Casselgren, Carl Johan
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Completing partial Latin squares with one filled row, column and symbol2013In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 313, no 9, p. 1011-1017Article in journal (Refereed)

Let P be an n x n partial Latin square every non-empty cell of which lies in a fixed row r, a fixed column c or contains a fixed symbols. Assume further that s is the symbol of cell (r, c) in P. We prove that P is completable to a Latin square if n >= 8 and n is divisible by 4, or n <= 7 and n is not an element of {3, 4, 5}. Moreover, we present a polynomial algorithm for the completion of such a partial Latin square. (C) 2013 Elsevier B.V. All rights reserved.

• 2. Cutler, Jonathan
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Cycle double covers of graphs with disconnected frames2004Report (Other academic)
• 3. Fleischner, Herbert
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Circuit double covers in special types of cubic graphs2009In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 309, no 18, p. 5724-5728Article in journal (Refereed)

Suppose that a 2-connected cubic graph G of order n has a circuit C of length at least n−4 such that GV(C) is connected. We show that G has a circuit double cover containing a prescribed set of circuits which satisfy certain conditions. It follows that hypohamiltonian cubic graphs (i.e., non-hamiltonian cubic graphs G such that Gv is hamiltonian for every vV(G)) have strong circuit double covers.

• 4.
Institute for Information Systems, Technical University of Vienna, Wien, Austria.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Cycle double covers containing certain circuits in cubic graphs having special structures2015In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 338, no 10, p. 1750-1754Article in journal (Refereed)

Our point of departure is Fleischner and Haggkvist (2014, Theorem 2). We first generalize this theorem. Then we apply it to cubic graphs whose vertex set can be decomposed into two classes, one class inducing a circuit and the other class inducing a (subdivision of a) caterpillar. (C) 2014 Elsevier B.V. All rights reserved.

• 5.
Vienna Univ Technol, Dept Data Bases & Artificial Intelligence, Inst Informat Syst, A-1040 Vienna, Austria.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Cycle double covers in cubic graphs having special structures2014In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 77, no 2, p. 158-170Article in journal (Refereed)

In the first part of this article, we employ Thomason's Lollipop Lemma [25] to prove that bridgeless cubic graphs containing a spanning lollipop admit a cycle double cover (CDC) containing the circuit in the lollipop; this implies, in particular, that bridgeless cubic graphs with a 2-factor F having two components admit CDCs containing any of the components in the 2-factor, although it need not have a CDC containing all of F. As another example consider a cubic bridgeless graph containing a 2-factor with three components, all induced circuits. In this case, two of the components may separately be used to start a CDC although it is uncertain whether the third component may be part of some CDC. Numerous other corollaries shall be given as well. In the second part of the article, we consider special types of bridgeless cubic graphs for which a prominent circuit can be shown to be included in a CDC. The interest here is the proof technique and therefore we only give the simplest case of the theorem. Notably, we show that a cubic graph that consists of an induced 2k-circuit C together with an induced 4k-circuit T and an independent set of 2k vertices, each joined by one edge to C and two edges to T, has a CDC starting with T.

• 6.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
• 7.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
A note on edge-decompositions of planar graphs2004In: Discrete Mathematics, ISSN 0012-365X, Vol. 283, no 1-3, p. 263-266Article in journal (Refereed)
• 8.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Condensed Matter Theory, Department of Physics, Royal Institute of Technology, SE-106 91, Stockholm, Sweden.
The Ising partition function for 2D grids with periodic boundary: computation and analysis2002In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 108, no 3-4, p. 429-457Article in journal (Refereed)

The Ising partition function for a graph counts the number of bipartitions of the vertices with given sizes, with a given size of the induced edge cut. Expressed as a 2-variable generating function it is easily translatable into the corresponding partition function studied in statistical physics. In the current paper a comparatively efficient transfer matrix method is described for computing the generating function for the n×n grid with periodic boundary. We have applied the method to up to the 15×15 grid, in total 225 vertices. We examine the phase transition that takes place when the edge cut reaches a certain critical size. From the physical partition function we extract quantities such as magnetisation and susceptibility and study their asymptotic behaviour at the critical temperature.

• 9.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Cycle double covers and spanning minors I2006In: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 96, no 2, p. 183-206Article in journal (Refereed)

Define a graph to be a Kotzig graph if it is $m$-regular and has an $m$-edge colouring in which each pair of colours form a Hamiltonian cycle. We show that every cubic graph with spanning subgraph consisting of a subdivision of a Kotzig graph together with even cycles has a cycle double cover, in fact a 6-CDC. We prove this for two other families of graphs similar to Kotzig graphs as well. In particular, let $F$ be a 2-factor in a cubic graph $G$ and denote by $G_{F}$ the pseudograph obtained by contracting each component in $F$. We show that if there exist a cycle in $G_{F}$ through all vertices of odd degree, then $G$ has a CDC. We conjecture that every 3-connected cubic graph contains a spanning subgraph homeomorphic to a Kotzig graph. In a sequel we show that every cubic graph with a spanning homeomorph of a 2-connected cubic graph on at most 10 vertices has a CDC.

• 10.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Cycle double covers and spanning minors II2006In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 306, no 8-9, p. 762-778Article in journal (Refereed)

In this paper we continue our investigations from [R. Häggkvist, K. Markström, Cycle double covers and spanning minors, Technical Report 07, Department of Mathematics, Umeå University, Sweden, 2001, J. Combin. Theory, Ser. B, to appear] regarding spanning subgraphs which imply the existence of cycle double covers. We prove that if a cubic graph G has a spanning subgraph isomorphic to a subdivision of a bridgeless cubic graph on at most 10 vertices then G has a CDC. A notable result is thus that a cubic graph with a spanning Petersen minor has a CDC, a result also obtained by Goddyn [L. Goddyn, Cycle covers of graphs, Ph.D. Thesis, University of Waterloo, 1988].

• 11.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden. Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Computation of the Ising partition function for two-dimensional square grids2004In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 69, no 4, p. 19-, article id 046104Article in journal (Refereed)

An improved method for obtaining the Ising partition function for $n \times n$ square grids with periodic boundary is presented. Our method applies results from Galois theory in order to split the computation into smaller parts and at the same time avoid the use of numerics. Using this method we have computed the exact partition function for the $320 \times 320$-grid, the $256 \times 256$-grid, and the $160 \times 160$-grid, as well as for a number of smaller grids. We obtain scaling parameters and compare with what theory prescribes.

• 12.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Physics, AlbaNova University Center, KTH, SE-106 91, Stockholm, Sweden . Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Physics, AlbaNova University Center, KTH, SE-106 91, Stockholm, Sweden; Department of Biosciences at Novum, Karolinska institutet, SE-141 57, Huddinge, Sweden. Department of Physics, AlbaNova University Center, KTH, SE-106 91, Stockholm, Sweden . Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A Monte Carlo sampling scheme for the Ising model2004In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 114, no 1-2, p. 455-480Article in journal (Refereed)

In this paper we describe a Monte Carlo sampling scheme for the Ising model and similar discrete-state models. The scheme does not involve any particular method of state generation but rather focuses on a new way of measuring and using the Monte Carlo data. We show how to reconstruct the entropy S of the model, from which, e.g., the free energy can be obtained. Furthermore we discuss how this scheme allows us to more or less completely remove the effects of critical fluctuations near the critical temperature and likewise how it reduces critical slowing down. This makes it possible to use simple state generation methods like the Metropolis algorithm also for large lattices.

• 13.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Ising model for the simple cubic lattice2007In: Advances in Physics, ISSN 0001-8732, E-ISSN 1460-6976, Vol. 56, no 5, p. 653-755Article, review/survey (Refereed)

The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.

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