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• 1. Aaghabali, M.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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, 132-144 p.Article in journal (Refereed)

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. Akbari, Saieed
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On 1-sum flows in undirected graphs2016In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 31, 646-665 p.Article in journal (Refereed)

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.

• 3.
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.
Avoiding Arrays of Odd Order by Latin Squares2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, 184-212 p.Article in journal (Refereed)

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.

• 4.
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)

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.

• 5.
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)

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.

• 6.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoidability of random arraysManuscript (preprint) (Other academic)

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 $K_{n,n}$ 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.

• 7.
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)

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.

• 8.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On Latin squares and avoidable arrays2010Doctoral thesis, comprehensive summary (Other academic)

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.

• 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. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding arrays of odd order by Latin squaresManuscript (preprint) (Other academic)

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.

• 10.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Southern Polytechnic State University, Marietta, Georgia. 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, 88-97 p.Article in journal (Refereed)

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.

• 11.
Christian-Albrechts Universität Kiel.
Christian-Albrechts Universität Kiel. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Christian-Albrechts Universität Kiel. Christian-Albrechts Universität Kiel.
Exact and heuristic algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection2015In: Journal of the Operational Research Society, ISSN 0160-5682, E-ISSN 1476-9360, Vol. 66, no 4, 615-626 p.Article in journal (Refereed)

We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.

• 12.
Christian-Albrechts Universität Kiel, Germany.
Christian-Albrechts Universität Kiel, Germany. Christian-Albrechts Universität Kiel, Germany.
Construction of Sparse Asymmetric Connectors2003In: Proceedings of European Conference on Combinatorics, Graph Theory and Applications (Eurocomb 2003), 2003Conference paper (Refereed)
• 13.
Christian-Albrechts-Universität Kiel, Germany.
Christian-Albrechts-Universität Kiel, Germany. Christian-Albrechts-Universität Kiel, Germany.
Constructions of Sparse Asymmetric Connectors2003In: Proceedings of 23rd Conference of Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2003) / [ed] P.K. Lodaya and J. Radhakrishnan, Berlin-Heidelberg: Springer Berlin/Heidelberg, 2003, 13-22 p.Conference paper (Refereed)
• 14.
Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany.
Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany. Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany.
Constructions of Sparse Asymmetric Connectors with Number Theoretic Methods2005In: Networks, ISSN 0028-3045, E-ISSN 1097-0037, Vol. 45, no 3, 119-124 p.Article in journal (Refereed)

We consider the problem of connecting a set I of n inputs to a set O of N outputs (n ≤ N) by as few edges as possible such that for every injective mapping f : I → O there are n vertex disjoint paths from i to f(i) of length k for a given k $\in \mathbb{N}$. For k = Ω(log N + log$^{2}$n) Oruς (1994) gave the presently best (n,N)-connector with O(N+n·log n) edges. For k=2 and N the square of a prime, Richards and Hwang (1985) described a construction using $N\left[\sqrt{n+5/4-1/2}\right] + n\left[\sqrt{n+5/4-1/2} \right]\sqrt{N}$edges. We show by a probabilistic argument that an optimal (n,N)-connector has Θ (N) edges, if $n\leq N^{1/2-\varepsilon}$for some ε>0. Moreover, we give explicit constructions based on a new number theoretic approach that need at most $N\left[\sqrt{3n/4} \right]+2n\left[\sqrt{3n/4} \right]\left[\sqrt{N} \right]$edges for arbitrary choices of n and N. The improvement we achieve is based on applying a generalization of the Erdös-Heilbronn conjecture on the size of restricted sums.

• 15.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Department of Information Science, Stellenbosch University, Stellenbosch, South Africa.
Minimization of Finite State Automata Through Partition Aggregation2016In: Logical Aspects of Computational Linguistics: Celebrating 20 Years of LACL (1996–2016) / [ed] Amblard, M DeGroote, P Pogodalla, S Retore, C, SPRINGER-VERLAG BERLIN , 2016, 328-328 p.Conference paper (Refereed)
• 16.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On avoiding some families of arrays2012In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 5, 963-972 p.Article in journal (Refereed)

An n×n array A with entries from {1,…,n} is avoidable if there is an n×n Latin square L such that no cell in L contains a symbol that occurs in the corresponding cell in A. We show that the problem of determining whether an array that contains at most two entries per cell is avoidable is NP-complete, even in the case when the array has entries from only two distinct symbols. Assuming that PNP, this disproves a conjecture by Öhman. Furthermore, we present several new families of avoidable arrays. In particular, every single entry array (arrays where each cell contains at most one symbol) of order n≥2k with entries from at most k distinct symbols and where each symbol occurs in at most n−2 cells is avoidable, and every single entry array of order n, where each of the symbols 1,…,n occurs in at most cells, is avoidable. Additionally, if k≥2, then every single entry array of order at least n≥4, where at most k rows contain non-empty cells and where each symbol occurs in at most nk+1 cells, is avoidable.

• 17.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On some graph coloring problems2011Doctoral thesis, comprehensive summary (Other academic)
• 18. Casselgren, Carl Johan
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Coloring Complete and Complete Bipartite Graphs from Random Lists2016In: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 32, no 2, 533-542 p.Article in journal (Refereed)

Assign to each vertex v of the complete graph on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f(n)-subsets of a color set , where f(n) is some integer-valued function of n. Such a list assignment L is called a random (f(n), [n])-list assignment. In this paper, we determine the asymptotic probability (as ) of the existence of a proper coloring of , such that for every vertex v of . We show that this property exhibits a sharp threshold at . Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph with parts of size m and n, respectively. We show that if , , and L is a random (f(n), [n])-list assignment for the line graph of , then with probability tending to 1, as , there is a proper coloring of the line graph of with colors from the lists.

• 19. 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, 1011-1017 p.Article 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.

• 20.
School of Computing and Mathematics, UCLan Cyprus, Pyla, Cyprus.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The thresholds for diameter 2 in random Cayley graphs2014In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 45, no 2, 218-235 p.Article in journal (Refereed)
• 21.
Umeå University, Faculty of Science and Technology, Department of mathematics. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA.
Unavoidable subgraphs of colored graphs2008In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 308, no 19, 4396-4413 p.Article in journal (Refereed)

A natural generalization of graph Ramsey theory is the study of unavoidable sub-graphs in large colored graphs. In this paper. we find a minimal family of unavoidable graphs in two-edge-colored graphs. Namely, for a positive even integer k, let y(k) be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint K-k/2's or simply one K-k/2. Bollobas conjectured that for all k and epsilon > 0, there exists an n(k, epsilon) such that if n >= n(k, epsilon) then every two-edge-coloring of K-n, in which the density of each color is at least epsilon, contains a member of this family. We solve this conjecture and present a series of results bounding it (k, s) for different ranges of epsilon. In particular, if epsilon is sufficiently close to 1/2, the gap between out upper and lower bounds for n(k, epsilon) is smaller than those for the classical Ramsey number R(k, k).

• 22. Cutler, Jonathan
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Latin squares with forbidden entries2004Report (Other academic)
• 23.
Austin Peay State University.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Extending partial latin cubes2014In: Ars combinatoria, ISSN 0381-7032, Vol. 113, 405-414 p.Article in journal (Refereed)

In the spirit of Ryser's theorem, we prove sufficient conditions on k, and m so that k xexm Latin boxes, i.e. partial Latin cubes whose filled cells form a k x x m rectangular box, can be extended to akxnxm latin box, and also to akxnxn latin box, where n is the number of symbols used, and likewise the order of the Latin cube. We also prove a partial Evans type result for Latin cubes, namely that any partial Latin cube of order n with at most n 1 filled cells is completable, given certain conditions on the spatial distribution of the filled cells.

• 24.
Computer Science Institute, University of Halle Wittenberg, Germany .
Computer Science Institute, University of Halle Wittenberg, Germany . Computer Science Institute, University of Halle Wittenberg, Germany . Computer Science Institute, University of Halle Wittenberg, Germany . Computer Science Institute, University of Halle Wittenberg, Germany .
Effective Heuristics for Large Euclidean TSP Instances Based on Pseudo Backbones2009In: Proceedings of 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2009) / [ed] Sonia Cafieri, Antonio Mucherino, Giacomo Nannicini, Fabien Tarissan and Leo Liberti, 2009, 3-6 p.Conference paper (Refereed)

We present two approaches for the Euclidean TSP which compute high quality tours for large instances. Both approaches are based on pseudo backbones consisting of all common edges of good tours. The first approach starts with some pre-computed good tours. Using this approach we found record tours for seven VLSI instances. The second approach is window based and constructs from scratch very good tours of huge TSPinstances, e. g., the World TSP.

• 25.
Computer Science Institute, University of Halle-Wittenberg, Germany.
Computer Science Institute, University of Halle-Wittenberg, Germany. Computer Science Institute, University of Halle-Wittenberg, Germany. Computer Science Institute, University of Halle-Wittenberg, Germany.
Effective Tour Searching for TSP by Contraction of Pseudo Backbone Edges2009In: Proceedings of 5th International Conference on Algorithmic Aspects in Information and Management (AAIM 2009) / [ed] A. Goldberg and Y. Zhou, Berlin-Heidelberg: Springer , 2009, 175-187 p.Conference paper (Refereed)

We introduce a reduction technique for the well-known TSP. The basic idea of the approach consists of transforming a TSP instance to another one with smaller size by contracting pseudo backbone edges computed in a preprocessing step, where pseudo backbone edges are edges which are likely to be in an optimal tour. A tour of the small instance can be re-transformed to a tour of the original instance. We experimentally investigated TSP benchmark instances by our reduction technique combined with the currently leading TSP heuristic of Helsgaun. The results strongly demonstrate the effectivity of this reduction technique: for the six VLSI instances xvb13584, pjh17845, fnc19402, ido21215, boa28924, and fht47608 we could set world records, i.e., find better tours than the effective reduction of the problem size so that we can search the more important tour subspace more intensively.

• 26.
Computer Science Institute, Christian-Albrechts-University of Kiel, Germany.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The b-Matching Problem in Hypergraphs: Hardness and Approximability2012In: Proceedings of 6th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2012) / [ed] Guohui Lin, Berlin-Heidelberg: Springer Berlin-Heidelberg , 2012, 200-211 p.Conference paper (Refereed)

In this paper we analyze the maximum cardinality $b$-matching problem in $l$-uniform hypergraphs with respect to the complexity class MAX-SNP, where $b$-matching is defined as follows: for given $b\in\mathbb{N}$ and a hypergraph $\mathcal{H}=(V,\varepsilon)$ a subset $M_{b}\subseteq\varepsilon$ with maximum cardinality is sought so that no vertex is contained in more than $b$ hyperedges of $M_{b}$. We show that if the maximum degree of the vertices is bounded by a constant $B\in\mathbb{N}$, this problem has no approximation scheme, unless $\mathcal{P}=\mathcal{N}\mathcal{P}$. This result generalizes a result of Kann from $b=1$ to the case that $b\in\mathbb{N}$ with $0. Furthermore, we extend a result of Srivastav and Stangier, who gave an approximation algorithm for the unweighted b-matching problem.

• 27.
Martin-Luther-University Halle-Wittenberg, Germany and GISA GmbH, D-06112 Halle (Saale), Germany.
Martin-Luther-University Halle-Wittenberg, Germany. Martin-Luther-University Halle-Wittenberg, Germany and Christian-Albrechts-University Kiel, Germany. Martin-Luther-University Halle-Wittenberg, Germany. Martin-Luther-University Halle-Wittenberg, Germany.
Finding Good Tours for Huge Euclidean TSP Instances by Iterative Backbone Contraction2010In: Proceedings of 6th International Conference on Algorithmic Aspects in Information and Management (AAIM 2010) / [ed] B. Chen, Berlin-Heidelberg: Springer , 2010, Vol. 6124, 119-130 p.Conference paper (Refereed)

This paper presents an iterative, highly parallelizable approach to find good tours for very large instances of the Euclidian version of the well-known Traveling Salesman Problem (TSP). The basic idea of the approach consists of iteratively transforming the TSP instance to another one with smaller size by contracting pseudo backbone edges. The iteration is stopped, if the new TSP instance is small enough for directly applying an exact algorithm or an efficient TSP heuristic. The pseudo backbone edges of each iteration are computed by a window based technique in which the TSP instance is tiled in non-disjoint sub-instances. For each of these sub-instances a good tour is computed, independently of the other sub-instances. An edge which is contained in the computed tour of every sub-instance (of the current iteration) containing this edge is denoted to be a pseudo backbone edge. Paths of pseudo-backbone edges are contracted to single edges which are fixed during the subsequent process.

• 28.
School of Mathematical Sciences Queen Mary University of London, London E1 4NS, UK.
Minimal weight in union-closed families2011In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 18, no 1, P95- p.Article in journal (Refereed)

Let Omega be a finite set and let S subset of P(Omega) be a set system on Omega. For x is an element of Omega, we denote by d(S)(x) the number of members of S containing x.Along-standing conjecture of Frankl states that if S is union-closed then there is some x is an element of Omega with d(S)(x)>= 1/2|S|. We consider a related question. Define the weight of a family S to be w(S) := A.S|A|.SupposeSisunion-closed. How small can w(S) be? Reimer showed w(S) >= 1/2|S|log(2)|S|, and that this inequality is tight. In this paper we show how Reimer's bound may be improved if we have some additional information about the domain Omega of S: if S separates the points of its domain, then w(S) >= ((vertical bar Omega vertical bar)(2)). This is stronger than Reimer's Theorem when |Omega| > root|S|log(2)|S|. In addition we constructa family of examples showing the combined bound on w(S)istightexcept in the region |Omega| = Theta(root|S|log(2)|S|), where it may be off by a multiplicative factor of 2. Our proof also gives a lower bound on the average degree: if S is a point-separating union-closed family on Omega, then 1/ |Omega|Sigma(x is an element of Omega)d(S)(x)>= 1/2 root|S|log(2)|S| broken vertical bar O(1), and this is best possible except for a multiplicative factor of 2.

• 29.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Sperner's Problem for G-Independent Families2015In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 24, no 3, 528-550 p.Article in journal (Refereed)

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edgeless graph, this problem is resolved by Sperner's theorem. In this paper, we focus on the case where G is the path of length n - 1, proving that the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).

• 30. Falgas-Ravry, Victor
Separating path systems2014In: Journal of Combinatorics, ISSN 2156-3527, E-ISSN 2150-959X, Vol. 5, no 3, 335-354 p.Article in journal (Refereed)

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every nn-vertex graph admits a separating path system of size linear in nn and we prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs with linear minimum degree. We also obtain tight bounds on the size of a minimal separating path system in the case of trees.

• 31. Falgas-Ravry, Victor
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.
Speed and concentration of the covering time for structured coupon collectorsManuscript (preprint) (Other academic)

Let be an n-set, and let be a random variable taking values in the powerset of V. Suppose we are given a sequence of random coupons X1,X2,…, where the Xi are independent random variables with distribution given by X. The covering time T is the smallest integer t≥0 such that ⋃ti=1Xi=V. The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focussed almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.

In this paper we study the covering time for much more general random variables X; we give general criteria for being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where fails to be concentrated and when structural properties in the distribution of allow for a very different behaviour of relative to the symmetric/uniform case.

• 32.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Codegree thresholds for covering 3-uniform hypergraphs2016In: SIAM Journal on Discrete Mathematics, ISSN 0895-4801, E-ISSN 1095-7146, Vol. 30, no 4, 1899-1917 p.Article in journal (Refereed)

Given two 3-uniform hypergraphs F and G = (V, E), we say that G has an F-covering if we can cover V with copies of F. The minimum codegree of G is the largest integer d such that every pair of vertices from V is contained in at least d triples from E. Define c(2)(n, F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F-covering. Determining c(2)(n, F) is a natural problem intermediate (but distinct) from the well-studied Turan problems and tiling problems. In this paper, we determine c(2)(n, K-4) (for n > 98) and the associated extremal configurations (for n > 998), where K-4 denotes the complete 3-graph on 4 vertices. We also obtain bounds on c(2)(n, F) which are apart by at most 2 in the cases where F is K-4(-) (K-4 with one edge removed), K-5(-), and the tight cycle C-5 on 5 vertices.

• 33. 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, 5724-5728 p.Article 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.

• 34.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA; Berlin Mathematical School, Berlin, Germany.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA . Department of Physics, AlbaNova University Center, KTH, 106 91, Stockholm, Sweden . Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Validations of the Asymptotic Matching Conjectures2008In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 133, no 3, 513-533 p.Article in journal (Refereed)

In this paper we review the asymptotic matching conjectures for r-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density p, for any p is an element of[0,1], for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite r-regular bipartite graphs.

• 35. Friedland, S.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the number of matching in regular graphs2008In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 15, no 1Article in journal (Refereed)

For the set of graphs with a given degree sequence, consisting of any number of 2's and 1's, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of m-matchings. We find the expected value of the number of m-matchings of r-regular bipartite graphs on 2n veritces with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for m-matchings in r-regular bipartite graphs on 2n vertices, and their asymptotic versions for infinite r-regular bipartite graphs. We prove these conjectures for 2-regular bipartite graphs and for m-matchings with m <= 4.

• 36. Friedland, S.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the number of matchings in regular graphs2008In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 15, no 1Article in journal (Refereed)
• 37.
Faculty to Economics Sciences, University of Groningen, the Netherlands and Department of Applied Mathematics, Khmelnitsky National University, Ukraine. Department of Computer Science, Royal Holloway University of London, UK and Department of Computer Science, University of Haifa, Israel. Computer Science Institute, University of Halle-Wittenberg, Germany.
Improving the Performance of Greedy Heuristics for TSPs Using Tolerances2007In: Communications in Dependability and Quality Management, Vol. 10, no 1, 52-70 p.Article in journal (Refereed)

In this paper we introduce three greedy algorithms for the traveling salesman problem. These algorithms are unique in that they use arc tolerances, rather than arc weights, to decide whether or not to include an arx in a solution. We report extensive computational experiments on benchmark instances that clearly demonstrate that our tolerance-based algorithms outperform their weight-based counterpart. Along with other papers dealing with the Assignment Problem, this paper indicates that the potential for using tolerance-based algorithms for various optimization problems is high and motivates further investigation of the approach. We recommend one of our algorithms as a significantly better alternative to the weight-based greedy, which is often used to produce initial TSP tours.

• 38.
Faculty of Economic Sciences, University of Groningen, The Netherlands. Department of Computer Science, Royal Holloway University of London, UK and Department of Computer Science, University of Haifa, Israel. Computer Science Institute, University of Halle-Wittenberg, Germany.
Tolerance-based algorithms for the traveling salesman problem2008In: Mathematical Programming and Game Theory for Decision Making / [ed] S.K. Neogy, R.B. Bapat, A.K. Das, and T. Parthasarathy, New Jersey: World Scientific, 2008, 47-59 p.Chapter in book (Refereed)

Most research on algorithms for combinatorial optimization use the costs of the elements in the ground set for making decisions about the solutions that the algorithms would output. For traveling salesman problems, this implies that algorithms generally use arc lengths to decide on whether an arc is included in a partial solution or not. In this paper we study the eect of using element tolerances for making these decisions. We choose the traveling salesman problem as a model combinatorial optimization problem and propose several greedy algorithms for it based on tolerances. We report extensive computational experiments on benchmark instances that clearly demonstrate that our tolerance-based algorithms outperform their weight-based counterpart. This indicates that the potential for using tolerance-based algorithms for various optimization problems is high and motivates further investigation of the approach.

• 39.
Christian-Albrechts Universität Kiel.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Christian-Albrechts Universität Kiel. Christian-Albrechts Universität Kiel.
Bounds for Static Black-Peg AB Mastermind2017In: Proceedings of 11th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2017), Springer Berlin/Heidelberg, 2017Conference paper (Refereed)

Mastermind is a famous two-player game introduced by M. Meirowitz (1970). Its combinatorics has gained increased interest over the last years   for different variants.

In this paper we consider a version known as the Black-Peg AB Game, where one player creates a secret code consisting of c colors on p <= c pegs, where each color is used at most once. The second player tries to guess the secret code with as few questions as possible. For each question he receives the number of correctly placed colors. In the static variant the second player doesn't receive the answers one at a time,  but all at once after asking the last question.  There are several results both for the AB and the static version, but the combination of both versions has not been considered so far. The most prominent case is n:=p=c, where the secret code and all questions are permutations. The main result of this paper is an upper bound of O(n^{1.525}) questions for this setting. With a slight modification of the arguments of Doerr et al. (2016) we also give a lower bound of \Omega(n\log n). Furthermore, we complement the upper bound for p=c  by an optimal (\lceil 4c/3 \rceil -1)-strategy for the special case p=2 and arbitrary c >= 2 and list optimal strategies for six additional pairs (p,c) .

• 40.
Department of Econometrics and Operations Research, University of Groningen, The Netherlands.
Computer Science Institute, Martin-Luther-University, Halle-Wittenberg, Germany.
How To Make a Greedy Heuristic for the Asymmetric Traveling Salesman Problem Competitive2005Report (Refereed)

It is widely confirmed by many computational experiments that a greedy type heuristics for the Traveling Salesman Problem (TSP) produces rather poor solutions except for the Euclidean TSP. The selection of arcs to be included by a greedy heuristic is usually done on the base of cost values. We propose to use upper tolerances of an optimal solution to one of the relaxed Asymmetric TSP (ATSP) to guide the selection of an arc to be included in the final greedy solution. Even though it needs time to calculate tolerances, our computational experiments for the wide range of ATSP instances show that tolerance based greedy heuristics is much more accurate and faster than previously reported greedy type algorithms.

• 41.
Faculty of Economics and Business, University of Groningen, The Netherlands.
Computer Science Institute, University of Halle-Wittenberg, D-06099 Halle (Saale), Germany.
The Computational Efficiency of Ji-Lee-Li Algorithm for the Assignment Problem2008In: Algorithmic Operations Research, ISSN 1718-3235, Vol. 3, no 1, 79-81 p.Article in journal (Refereed)

Ji et al. have conjectured that using the matrix form (to represent a basic solution) instead of the Simplex tableau in the dual Simplex method will lead to an algorithm with the time complexity comparable to the Hungarian algorithm for solving the Assignment Problem. In this note we show that both the time complexity and the CPU times of the Ji et al. algorithm are far away from being competitive to the Hungarian algorithm.

• 42.
Faculty of Economic Sciences, University of Groningen, The Netherlands and Department of Applied Mathematics, Khmelnitsky National University, Ukraine.
University of Halle-Wittenberg, Computer Science Institute, Germany. University of Halle-Wittenberg, Computer Science Institute, Germany.
Some Basics on Tolerances2006In: Proceedings of 2nd International Conference on Alghorithmic Aspects in Information and Management (AAIM 2006) / [ed] S.-W. Cheng and C.K. Poon, Berlin-Heidelberg: Springer Berlin/Heidelberg, 2006, 194-206 p.Conference paper (Refereed)

In this paper we deal with sensitivity analysis of combinatorial optimization problems and its fundamental term, the tolerance. For three classes of objective functions (Σ,∏,MAX) we give some basic properties on upper and lower tolerances. We show that the upper tolerance of an element is well defined, how to compute the upper tolerance of an element, and give equivalent formulations when the upper tolerance is +∞ or > 0. Analogous results are given for the lower tolerance and some results on the relationship between lower and upper tolerances are given.

• 43.
Faculty of Economic Sciences, University of Groningen, The Netherlands and Department of Applied Mathematics, Khmelnitsky National University, Ukraine.
Computer Science Institute, University of Halle-Wittenberg, Germany. Computer Science Institute, University of Halle-Wittenberg, Germany.
Tolerance Based Contract-or-Patch Heuristic for the Asymmetric TSP2006In: Proceedings of 3rd Workshop on Combinatorial and Algorithmic Aspects of Networking (CAAN 2006) / [ed] T. Erlebach, Berlin-Heidelberg: Springer Berlin/Heidelberg, 2006, 86-97 p.Conference paper (Refereed)

In this paper we improve the quality of a recently suggested class of construction heuristics for the Asymmetric Traveling Salesman Problem (ATSP), namely the Contract-or-Patch heuristic. Our improvement is based on replacing the selection of each path to be contracted after deleting a heaviest arc from each short cycle in an Optimal Assignment Problem Solution (OAPS) by contracting a single arc from a short cycle in an OAPS with the largest upper tolerance with respect to one of the relaxed ATSP. The improved algorithm produces higher-quality tours than all previous COP versions and is clearly outperforming all other construction heuristics on robustness.

• 44.
Faculty of Economic Sciences, University of Groningen, The Netherlands and Department of Applied Mathematics, Khmelnitsky National University, Ukraine.
Computer Science Institute, University of Halle-Wittenberg, Germany. Computer Science Institute, University of Halle-Wittenberg, Germany.
Tolerances Applied in Combinatorial Optimization2006In: Journal of Computer Science, ISSN 1549-3636, E-ISSN 1552-6607, Vol. 2, no 9, 716-734 p.Article in journal (Refereed)

In this paper we deal with sensitivity analysis of combinatorial optimization problems and its fundamental term, the tolerance. For three classes of objective functions (Σ,II,MAX) we give some basic properties on upper and lower tolerances. We show that the upper tolerance of an element is well defined, how to compute the upper tolerance of an element, and give equivalent formulations when the upper tolerance is +∞ or > 0. Analogous results are given for the lower tolerance and some results on the relationship between lower and upper tolerances are given.

• 45.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On snarks that are far from being 3-edge colorable2016In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 23, no 2, P2.6Article in journal (Refereed)

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.

• 46.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On snarks that are far from being 3-edge colorableManuscript (preprint) (Other academic)

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices.     Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.

• 47.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Snarks: Generation, coverings and colourings2012Doctoral thesis, comprehensive summary (Other academic)

For a number of unsolved problems in graph theory such as the cycle double cover conjecture, Fulkerson's conjecture and Tutte's 5-flow conjecture it is sufficient to prove them for a family of graphs called snarks. Named after the mysterious creature in Lewis Carroll's poem, a \emph{snark} is a cyclically 4-edge connected 3-regular graph of girth at least 5 which cannot be properly edge coloured using three colours. Snarks and problems for which an edge minimal counterexample must be a snark are the central topics of this thesis.

The first part of this thesis is intended as a short introduction to the area. The second part is an introduction to the appended papers and the third part consists of the four papers presented in a chronological order.

In Paper I we study the strong cycle double cover conjecture and stable cycles for small snarks. We prove that if a bridgeless cubic graph $G$ has a cycle of length at least $|V(G)|-9$ then it also has a cycle double cover. Furthermore we show that there exist cyclically 5-edge connected snarks with stable cycles and that there exists an infinite family of snarks with stable cycles of length 24.

In Paper II we present a new algorithm for generating all non-isomorphic snarks with a given number of vertices. We generate all snarks on 36 vertices and less and study these with respect to various properties. We find that a number of conjectures on cycle covers and colourings holds for all graphs of these orders. Furthermore we present counterexamples to no less than eight published conjectures on cycle coverings, cycle decompositions and the general structure of regular graphs.

In Paper III we show that Jaeger's Petersen colouring conjecture holds for three infinite families of snarks and that a minimum counterexample to this conjecture cannot contain a certain subdivision of $K_{3,3}$ as a subgraph. Furthermore, it is shown that one infinite family of snarks have strong Petersen colourings while another does not have any such colourings.

Two simple constructions for snarks with arbitrary high oddness and resistance is given in Paper IV. It is observed that some snarks obtained from this construction have the property that they require at least five perfect matchings to cover the edges. This disproves a suggested strengthening of Fulkerson's conjecture.

• 48.
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 stable cycles and cycle double covers of graphs with large circumference2012In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 17, 2540-2544 p.Article in journal (Refereed)

A cycle C in a graph is called stable if there exists no other cycle D in the same graph such that V(C)⊆V(D). In this paper, we study stable cycles in snarks and we show that if a cubic graph G has a cycle of length at least |V(G)|−9 then it has a cycle double cover. We also give a construction for an infinite snark family with stable cycles of constant length and answer a question by Kochol by giving examples of cyclically 5-edge connected snarks with stable cycles.

• 49.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Petersen-colorings and some families of snarks2014In: Ars Mathematica Contemporanea, ISSN 1855-3966, Vol. 7, no 1, 161-173 p.Article in journal (Other academic)

In this paper we study Petersen-colorings and strong Petersen-colorings on some well known families of snarks, e.g. Blanusa snarks, Goldberg snarks and flower snarks. In particular, it is shown that flower snarks have a Petersen-coloring but they do not have a strong Petersen-coloring. Furthermore it is proved that possible minimum counterexamples to Jaeger's Petersen-coloring conjecture do not contain a specific subdivision of K-3,K-3.

• 50.
Queen Mary Univ London, Sch Math Sci, London E1 4NS, England.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Turán and Ramsey properties of subcube intersection graphs2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 1, 55-70 p.Article in journal (Refereed)

The discrete cube {0, 1}d is a fundamental combinatorial structure. A subcube of {0, 1}d is a subset of 2k of its points formed by fixing k coordinates and allowing the remaining d - k to vary freely. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no r + 1 of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no k which have non-empty intersection and no l which are pairwise disjoint? These questions are naturally expressed using intersection graphs. The intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Let I(n, d) be the set of all n vertex graphs which can be represented as the intersection graphs of subcubes in {0, 1}d. With this notation our first question above asks for the largest number of edges in a Kr+1-free graph in I(n, d). As such it is a Turán-type problem. We answer this question asymptotically for some ranges of r and d. More precisely we show that if (k + 1)2 [d/k+1] &lt; n ≥k2[d/k] for some integer k ≥ 2 then the maximum edge density is (1 - 1/k - o(1)) provided that n is not too close to the lower limit of the range. The second question can be thought of as a Ramsey-type problem. The maximum such n can be defined in the same way as the usual Ramsey number but only considering graphs which are in I(n, d). We give bounds for this maximum n mainly concentrating on the case that l is fixed, and make some comparisons with the usual Ramsey number.

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