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Pham, Lan Anh

Open this publication in new window or tab >>Latin cubes with forbidden entries### Casselgren, Carl Johan

### Markström, Klas

### Pham, Lan Anh

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}); 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.

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 >>On avoiding and completing colorings### Pham, Lan Anh

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2019 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Umeå: Umeå University, 2019. p. 11
##### Series

Research report in mathematics, ISSN 1653-0810 ; 66/19
##### Keywords

Graph coloring, precoloring, list assignment
##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-158343 (URN)978-91-7855-055-5 (ISBN)
##### Public defence

2019-05-24, MA121, MIT-huset, Umeå, 10:15 (English)
##### Opponent

### Eriksson, Kimmo

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_j_idt359",{id:"formSmash:j_idt184:1:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_j_idt359",multiple:true});
##### Supervisors

### Markström, Klas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Casselgren, Carl Johan

### Falgas-Ravry, Victor

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

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_j_idt365",{id:"formSmash:j_idt184:1:j_idt188:j_idt365",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_j_idt365",multiple:true});
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Available from: 2019-05-03 Created: 2019-04-24 Last updated: 2019-04-29Bibliographically approved

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

All of my papers are related to the problem of avoiding and completing an edge precoloring of a graph. In more detail, given a graph G and a partial proper edge precoloring φ of G and a list assignment L for every non-colored edge of G, can we extend φ to a proper edge coloring of G which avoids L?

In Paper I, G is the d-dimensional hypercube graph Q_{d}, a partial proper edge precoloring φ and a list assignment L must satisfy certain sparsity conditions. Paper II still deals with the hypercube graph Q_{d}, but the list assignment L for every edge of Q_{d} is an empty set and φ must be a partial proper edge precoloring of at most d-1 edges. In Paper III, G is a (d,s)-edge colorable graph; that is G has a proper d-edge coloring, where every edge is contained in at least s-1 2-colored 4-cycles, L must satisfy certain sparsity conditions and we do not have a partial proper edge precoloring φ on edges of G. The problem in Paper III is also considered in Paper IV and Paper V, but here G can be seen as the complete 3-uniform 3-partite hypergraph K^{3}_{n,n,n}, where n is a power of two in paper IV and n is an even number in paper V.

Mälardalens högskola, Sverige.

Department of Mathematics (MAI), Linköping University, Sweden.

Open this publication in new window or tab >>chi-bounds, operations, and chords### Trotignon, Nicolas

### Pham, Lan Anh

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}); 2018 (English)In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 88, no 2, p. 312-336Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

chi-bounded, amalgam, chords
##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-147426 (URN)10.1002/jgt.22214 (DOI)000430108800007 ()
#####

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Available from: 2018-07-20 Created: 2018-07-20 Last updated: 2018-07-20Bibliographically approved

A long unichord in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is long unichord free if it does not contain any long unichord. We prove a structure theorem for long unichord free graph. We give an O(n4m) time algorithm to recognize them. We show that any long unichord free graph G can be colored with at most O(3) colors, where is the maximum number of pairwise adjacent vertices in G.

Open this publication in new window or tab >>On avoiding and completing edge colorings### Pham, Lan Anh

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}); 2018 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Umeå: Umeå University, 2018. p. 8
##### Series

Research report in mathematics, ISSN 1653-0810
##### National Category

Discrete Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-146915 (URN)978-91-7601-876-7 (ISBN)
##### Presentation

2018-05-17, MA346, MIT Building, Umea University, Umea, 13:00 (English)
##### Opponent

### Linusson, Svante

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_j_idt359",{id:"formSmash:j_idt184:3:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_j_idt359",multiple:true});
##### Supervisors

### Markström, Klas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_j_idt365",{id:"formSmash:j_idt184:3:j_idt188:j_idt365",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_j_idt365",multiple:true});
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Available from: 2018-05-25 Created: 2018-05-04 Last updated: 2018-06-09Bibliographically approved

These papers are all related to the problem of avoiding and completing an edge precoloring of a graph. In more detail, given a graph G and a partial proper edge precoloring φ of G and a list assignment L for every non-colored edge of G, can we extend the precoloring to a proper edge coloring avoiding any list assignment? In the first paper, G is a d-dimensional hypercube graph Q_{d}, a partial proper edge precoloring φ and every list assignment L must satisfy certain sparsity conditions. The second paper still deals with d-dimensional hypercube graph Q_{d}, but the list assignment L for every edge of Q_{d} is an empty set and φ must be a partial proper edge precoloring of at most (d - 1) edges. For the third paper, G can be seen as a complete 3-uniform 3-partite hypergraph, every list assignment L must satisfy certain sparsity conditions but we do not have a partial proper edge precoloring φ on edges of G.

Department of Mathematics, KTH Royal Institute of Technology.

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

### Markström, Klas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Pham, Lan Anh

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}); (English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### National Category

Discrete Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-147508 (URN)
#####

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Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2019-04-29

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}.

Open this publication in new window or tab >>Latin cubes of even order with forbidden entries### Casselgren, Carl Johan

### Pham, Lan Anh

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}); (English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### Keywords

Latin cubes, forbidden entries
##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-158322 (URN)
#####

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Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-04-29

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 γ>0 such that if n=2t 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.

Open this publication in new window or tab >>On restricted colorings of (d,s)-edge colorable graphs### Pham, Lan Anh

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.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}); (English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### Keywords

list assigment, restricted colorings
##### National Category

Discrete Mathematics
##### Identifiers

urn:nbn:se:umu:diva-158320 (URN)
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Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-04-29

A cycle is 2-colored if its edges are properly colored by two distinct colors. A (d,s)-edge colorable graph G is a d-regular graph that admits a proper d-edge coloring in which every edge of G is in at least s−1 2-colored 4-cycles. Given a (d,s)-edge colorable graph G and a list assigment L of forbidden colors for the edges of G satisfying certain sparsity conditions, we prove that there is a proper d-edge coloring of Gthat avoids L, that is, a proper edge coloring φ of G such that φ(e)∉L(e) for every edge e of G.

Open this publication in new window or tab >>Restricted extension of sparse partial edge colorings of hypercubes### Casselgren, Carl Johan

### Markström, Klas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Pham, Lan Anh

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}); (English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### National Category

Discrete Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-146922 (URN)
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Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2019-04-29

We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Q_{d}, and lists of allowed colors for the non-colored edges of Q_{d}, can we extend the partial coloring to a proper d-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.