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Casselgren, Carl Johan

Open this publication in new window or tab >>Avoiding Arrays of Odd Order by Latin Squares### Andren, Lina J.

### Casselgren, Carl Johan

### Öhman, Lars-Daniel

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2013 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, p. 184-212Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-66768 (URN)10.1017/S0963548312000570 (DOI)000314296400002 ()
#####

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Available from: 2013-03-15 Created: 2013-03-05 Last updated: 2018-06-08Bibliographically approved

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 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.

Open this publication in new window or tab >>Coloring graphs from random lists of size 2### Casselgren, Carl Johan

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##### Abstract [en]

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

London: Academic Press, 2012
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-50912 (URN)10.1016/j.ejc.2011.09.040 (DOI)000297890300005 ()
#####

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Available from: 2012-01-02 Created: 2012-01-02 Last updated: 2018-06-08Bibliographically approved

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

Let G = G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Delta. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set e of size sigma (n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring phi such that phi(v) is an element of L(v) for all v is an element of V(G). In particular, we show that if g is odd and sigma (n) = omega(n(1/(2g-2))), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n --> infinity. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n >= g, there is a graph H = H(n, g) with bounded maximum degree and girth g, such that if sigma (n) = 0(n(1/(2g-2))), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n --> infinity. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size sigma (n), exhibits a sharp threshold at sigma (n) = 2n. (C) 2011 Elsevier Ltd. All rights reserved.

Open this publication in new window or tab >>On avoiding some families of arrays### Casselgren, Carl Johan

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2012 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 5, p. 963-972Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Latin square, avoiding arrays, list coloring
##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-43325 (URN)10.1016/j.disc.2011.10.028 (DOI)
#####

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Available from: 2011-04-28 Created: 2011-04-27 Last updated: 2018-06-08Bibliographically approved

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 *P*≠*NP*, 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*≥2*k* 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 *n*−*k*+1 cells, is avoidable.

Open this publication in new window or tab >>A note on path factors of (3,4)-biregular bipartite graphs### Casselgren, Carl Johan

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2011 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 18, no 1, p. P218-Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

path factor, biregular bipartite graph, interval edge coloring
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-50410 (URN)000296974200001 ()
#####

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Available from: 2011-12-13 Created: 2011-12-08 Last updated: 2018-06-08Bibliographically approved

A proper edge coloring of a graph G with colors 1,2,3, ... is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graphis (3,4)-biregular if all vertices in one part have degree 3 and all vertices in the other part have degree 4. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph G has a spanning subgraph whose components are paths with end points at 3-valent vertices and lengths in {2,4,6,8}, then G has an interval coloring. It was also conjectured that every simple (3,4)-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple (3,4)-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at 3-valent vertices and lengths not exceeding 22.

Open this publication in new window or tab >>On some graph coloring problems### Casselgren, Carl Johan

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Place, publisher, year, edition, pages

Umeå: Umeå universitet, Institutionen för matematik och matematisk statistik, 2011. p. 22
##### Series

Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 48
##### Keywords

List coloring, interval edge coloring, coloring graphs from random lists, biregular graph, avoiding arrays, Latin square, scheduling
##### National Category

Discrete Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-43389 (URN)978-91-7459-198-9 (ISBN)
##### Public defence

2011-05-20, MIT-huset, MA121, Umeå universitet, Umeå, 13:15
##### Opponent

### Linusson, Svante

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_j_idt359",{id:"formSmash:j_idt184:4:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_j_idt359",multiple:true});
##### Supervisors

### Häggkvist, Roland

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_j_idt365",{id:"formSmash:j_idt184:4:j_idt188:j_idt365",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_j_idt365",multiple:true});
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Available from: 2011-04-29 Created: 2011-04-28 Last updated: 2018-06-08Bibliographically approved

Institutionen för Matematik, Kungliga Tekniska Högskolan, Stockholm.

Open this publication in new window or tab >>Vertex coloring complete multipartite graphs from random lists of size 2### Casselgren, Carl Johan

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2011 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 13, p. 1150-1157Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

List coloring; Complete multipartite graph; Random list
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-43323 (URN)10.1016/j.disc.2010.07.013 (DOI)
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Available from: 2011-04-27 Created: 2011-04-27 Last updated: 2018-06-08Bibliographically approved

Let *K*_{s×m} be the complete multipartite graph with *s* parts and *m* vertices in each part. Assign to each vertex *v* of *K*_{s×m} a list *L*(*v*) of colors, by choosing each list uniformly at random from all 2-subsets of a color set *C* of size *σ*(*m*). In this paper we determine, for all fixed *s* and growing *m*, the asymptotic probability of the existence of a proper coloring *φ*, such that *φ*(*v*)∈*L*(*v*) for all *v*∈*V*(*K*_{s×m}). We show that this property exhibits a sharp threshold at *σ*(*m*)=2(*s*−1)*m*.

Open this publication in new window or tab >>Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs### Asratian, Armen S.

### Casselgren, Carl Johan

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

### West, Douglas B.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2009 (English)In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 61, no 2, p. 88-97Article in journal (Refereed) Published
##### Abstract [en]

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

Wiley Periodicals Inc., 2009
##### Keywords

path factor, interval edge-coloring, biregular bipartite graph
##### National Category

Discrete Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-25912 (URN)10.1002/jgt.20370 (DOI)
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Available from: 2009-09-10 Created: 2009-09-10 Last updated: 2018-06-08Bibliographically approved

Linköping University, Linköping, Sweden.

Southern Polytechnic State University, Marietta, Georgia.

University of Illinois, Urbana, Illinois.

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.

Open this publication in new window or tab >>On Path Factors of (3,4)-Biregular Bigraphs### Asratian, Armen S.

### Casselgren, Carl Johan

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2008 (English)In: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 24, no 5, p. 405-411Article in journal (Refereed) Published
##### Abstract [en]

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

SpringerLink, 2008
##### Keywords

Path factor, Biregular bigraph, Interval edge coloring
##### Identifiers

urn:nbn:se:umu:diva-11276 (URN)10.1007/s00373-008-0803-y (DOI)
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Available from: 2008-12-05 Created: 2008-12-05 Last updated: 2018-06-09Bibliographically approved

Linköping University Linköping Sweden.

A (3, 4)-biregular bigraph *G* is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of *G* is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.

Open this publication in new window or tab >>On interval edge colorings of (a,b)-biregular bipartitie graphs### Asratian, Armen S.

### Casselgren, Carl Johan

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2006 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 307, no 15, p. 1951-1956Article in journal (Refereed) Published
##### Place, publisher, year, edition, pages

Elsevier B.V., 2006
##### Identifiers

urn:nbn:se:umu:diva-7867 (URN)10.1016/j.disc.2006.11.001 (DOI)
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Available from: 2008-01-13 Created: 2008-01-13 Last updated: 2018-06-09Bibliographically approved

Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden.

Open this publication in new window or tab >>A note on path factors of (3,4)-biregular bipartite graphs### Casselgren, Carl Johan

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); (English)Manuscript (preprint) (Other academic)
##### Identifiers

urn:nbn:se:umu:diva-43326 (URN)
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Available from: 2011-04-28 Created: 2011-04-27 Last updated: 2018-06-08Bibliographically approved