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Markström, Klas

Open this publication in new window or tab >>Edge precoloring extension of hypercubes### Casselgren, Carl Johan

### Markström, Klas

### Pham, Lan Anh

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##### Abstract [en]

##### Place, publisher, year, edition, pages

John Wiley & Sons, 2020
##### Keywords

edge coloring, hypercube, precoloring extension
##### National Category

Discrete Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-147508 (URN)10.1002/jgt.22561 (DOI)000520734800001 ()
#####

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##### Note

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.

We consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most d − 1 edges of the d ‐dimensional hypercube Q d can be extended to a proper d ‐edge coloring of Q d . Additionally, we characterize which partial edge colorings of Q d with precisely d precolored edges are extendable to proper d ‐edge colorings of Q d.

Originally included in thesis in manuscript form.

Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2020-05-13Bibliographically approvedOpen this publication in new window or tab >>Latin cubes with forbidden entries### Casselgren, Carl Johan

### Markström, Klas

### Pham, Lan Anh

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_1_j_idt195_some",{id:"formSmash:j_idt191:1:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_1_j_idt195_otherAuthors",{id:"formSmash:j_idt191:1:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 26, no 1, article id P1.2Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Newark: Department of Mathematical Science, University of Delaware, 2019
##### National Category

Discrete Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-147511 (URN)000456790800002 ()
#####

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##### Funder

Swedish Research Council, 2014-4897
##### Note

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant y>0 such that if n=2^{k} and A is a 3-dimensional n×n×n array where every cell contains at most γn symbols, and every symbol occurs at most γn times in every line of A, then A is avoidable; that is, there is a Latin cube L of order n such that for every 1 ≤ i,j,k ≤ n, the symbol in position (i,j,k) of L does not appear in the corresponding cell of A.

Originally included in thesis in manuscript form.

Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2019-04-24Bibliographically approvedOpen this publication in new window or tab >>Restricted completion of sparse partial Latin squares### Andren, Lina J.

### Casselgren, Carl Johan

### Markström, Klas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_some",{id:"formSmash:j_idt191:2:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_otherAuthors",{id:"formSmash:j_idt191:2:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 28, no 5, p. 675-695Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Cambridge University Press, 2019
##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-166828 (URN)10.1017/S096354831800055X (DOI)000500255000002 ()
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Available from: 2020-01-03 Created: 2020-01-03 Last updated: 2020-01-03Bibliographically approved

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.

Open this publication in new window or tab >>Revisiting the cavity-method threshold for random 3-SAT### Lundow, Per-Håkan

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.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_some",{id:"formSmash:j_idt191:3:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_otherAuthors",{id:"formSmash:j_idt191:3:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: Physical review. E, ISSN 2470-0045, E-ISSN 2470-0053, Vol. 99, no 2, article id 022106Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Physical Society, 2019
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:umu:diva-162509 (URN)10.1103/PhysRevE.99.022106 (DOI)000458140700001 ()30934345 (PubMedID)
#####

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Available from: 2019-08-21 Created: 2019-08-21 Last updated: 2019-08-21Bibliographically approved

A detailed Monte Carlo study of the satisfiability threshold for random 3-SAT has been undertaken. In combination with a monotonicity assumption we find that the threshold for random 3-SAT satisfies α_{3}≤4.262. If the assumption is correct, this means that the actual threshold value for *k*=3 is lower than that given by the cavity method. In contrast the latter has recently been shown to give the correct value for large *k*. Our result thus indicate that there are distinct behaviors for *k* above and below some critical *k _{c}*, and the cavity method may provide a correct mean-field picture for the range above

Open this publication in new window or tab >>Triples of Orthogonal Latin and Youden Rectangles of small order### Jäger, Gerold

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.### Öhman, Lars-Daniel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Shcherbak, Denys

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_some",{id:"formSmash:j_idt191:4:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_otherAuthors",{id:"formSmash:j_idt191:4:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 27, no 4, p. 229-250Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-158857 (URN)10.1002/jcd.21642 (DOI)000459040800001 ()2-s2.0-85059030594 (PubMedID)
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Available from: 2019-05-13 Created: 2019-05-13 Last updated: 2019-05-23Bibliographically approved

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal k × n Latin rectangles for k ≤ n ≤ 7. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of k×8 rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group is non-trivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of 4 × 8 rectangles, formed by juxtaposing two selected copies of complete sets of MOLS of order 4.

Open this publication in new window or tab >>Full subgraphs### Falgas-Ravry, Victor

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.### Verstraëte, Jacques

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_some",{id:"formSmash:j_idt191:5:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_otherAuthors",{id:"formSmash:j_idt191:5:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_otherAuthors",multiple:true}); 2018 (English)In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 88, no 3, p. 411-427Article in journal (Refereed) Published
##### Keywords

extremal graph theory, graph discrepancy, graph partitions
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-144161 (URN)10.1002/jgt.22221 (DOI)000432013200004 ()2-s2.0-85034034448 (Scopus ID)
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##### Funder

Swedish Research Council
Available from: 2018-01-23 Created: 2018-01-23 Last updated: 2018-06-13Bibliographically approved

University of California-San Diego.

Open this publication in new window or tab >>On 1-sum flows in undirected graphs### Akbari, Saieed

### Friedland, Shmuel

### Markström, Klas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Zare, Sanaz

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_some",{id:"formSmash:j_idt191:6:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_otherAuthors",{id:"formSmash:j_idt191:6:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_otherAuthors",multiple:true}); 2016 (English)In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 31, p. 646-665Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

INT LINEAR ALGEBRA SOC, 2016
##### Keywords

L-Flow, gamma-L-Flow, c-Sum flow, Bipartite graph
##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-134293 (URN)10.13001/1081-3810.3003 (DOI)000396550500002 ()
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Available from: 2017-05-09 Created: 2017-05-09 Last updated: 2018-06-09Bibliographically approved

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.

Open this publication in new window or tab >>Random subcube intersection graphs I: cliques and covering### Falgas-Ravry, Victor

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.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_some",{id:"formSmash:j_idt191:7:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_otherAuthors",{id:"formSmash:j_idt191:7:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_otherAuthors",multiple:true}); 2016 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 23, no 3, article id P3.43Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Random graphs, Random intersection graphs
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-127244 (URN)000385228700002 ()
#####

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Available from: 2016-11-14 Created: 2016-11-03 Last updated: 2018-06-09Bibliographically approved

We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube Q_{d} to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their intersection is non-empty. Our motivation for considering such graphs is to model 'random compatibility' between vertices in a large network. For both of the models considered in this paper, we determine the thresholds for covering the underlying hypercube Q_{d} and for the appearance of s-cliques. In addition we pose a number of open problems.

Open this publication in new window or tab >>The scaling window of the 5D Ising model with free boundary conditions### Lundow, Per-Håkan

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.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_some",{id:"formSmash:j_idt191:8:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_otherAuthors",{id:"formSmash:j_idt191:8:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_otherAuthors",multiple:true}); 2016 (English)In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 911, p. 163-172Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2016
##### National Category

Atom and Molecular Physics and Optics Probability Theory and Statistics
##### Identifiers

urn:nbn:se:umu:diva-127239 (URN)10.1016/j.nuclphysb.2016.08.003 (DOI)000384565800007 ()
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Available from: 2016-11-14 Created: 2016-11-03 Last updated: 2018-06-09Bibliographically approved

The five-dimensional Ising model with free boundary conditions has recently received a renewed interest in a debate concerning the finite-size scaling of the susceptibility near the critical temperature. We provide evidence in favour of the conventional scaling picture, where the susceptibility scales as L-2 inside a critical scaling window of width 1/L-2. Our results are based on Monte Carlo data gathered on system sizes up to L = 79(ca. three billion spins) for a wide range of temperatures near the critical point. We analyse the magnetisation distribution, the susceptibility and also the scaling and distribution of the size of the Fortuin-Kasteleyn cluster containing the origin. The probability of this cluster reaching the boundary determines the correlation length, and its behaviour agrees with the mean field critical exponent delta = 3, that the scaling window has width 1/L-2.

Open this publication in new window or tab >>Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary### Lundow, Per-Håkan

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.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_9_j_idt195_some",{id:"formSmash:j_idt191:9:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_9_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_9_j_idt195_otherAuthors",{id:"formSmash:j_idt191:9:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_9_j_idt195_otherAuthors",multiple:true}); 2015 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 91, article id 022112Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Physical Society, 2015
##### National Category

Condensed Matter Physics
##### Identifiers

urn:nbn:se:umu:diva-99987 (URN)10.1103/PhysRevE.91.022112 (DOI)000349910000001 ()
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Available from: 2015-02-17 Created: 2015-02-17 Last updated: 2018-06-07Bibliographically approved

The finite-size scaling behavior for the Ising model in five dimensions, with either free or cyclic boundary, has been the subject of a long-running debate. The older papers have been based on ideas from, e.g., field theory or renormalization. In this paper we propose a detailed and exact scaling picture for critical region of the model with cyclic boundary. Unlike the previous papers our approach is based on a comparison with the existing exact and rigorous results for the FK-random-cluster model on a complete graph. Based on those results we predict several distinct scaling regions in an L -dependent window around the critical point. We test these predictions by comparing with data from Monte Carlo simulations and find a good agreement. The main feature which differs between the complete graph and the five-dimensional model with free boundary is the existence of a bimodal energy distribution near the critical point in the latter. This feature was found by the same authors in an earlier paper in the form of a quasi-first-order phase transition for the same Ising model.