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On avoiding and completing edge colorings
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (Discrete Mathematics)
2018 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

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 Qd, 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 Qd, but the list assignment L for every edge of Qd 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. 

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: urn:nbn:se:umu:diva-146915ISBN: 978-91-7601-876-7 (print)OAI: oai:DiVA.org:umu-146915DiVA, id: diva2:1203795
Presentation
2018-05-17, MA346, MIT Building, Umea University, Umea, 13:00 (English)
Opponent
Supervisors
Available from: 2018-05-25 Created: 2018-05-04 Last updated: 2018-06-09Bibliographically approved
List of papers
1. Restricted extension of sparse partial edge colorings of hypercubes
Open this publication in new window or tab >>Restricted extension of sparse partial edge colorings of hypercubes
(English)In: Article in journal (Refereed) Submitted
Abstract [en]

We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Qd, and lists of allowed colors for the non-colored edges of Qd, 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.

National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-146922 (URN)
Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2018-06-09
2. Edge precoloring extension of hypercubes
Open this publication in new window or tab >>Edge precoloring extension of hypercubes
(English)In: Article in journal (Refereed) Submitted
Abstract [en]

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 Qd can be extended to a proper d-edge coloring of  Qd. Additionally, we characterize which partial edge colorings of  Qd with precisely d precolored edges are extendable to proper d-edge colorings of  Qd.

National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-147508 (URN)
Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2018-06-09
3. Latin cubes with forbidden entries
Open this publication in new window or tab >>Latin cubes with forbidden entries
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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=2k 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. 

National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-147511 (URN)
Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2018-06-09

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Pham, Lan Anh

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