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Restricted extension of sparse partial edge colorings of hypercubes
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
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.

Nationell ämneskategori
Diskret matematik
Forskningsämne
matematik
Identifikatorer
URN: urn:nbn:se:umu:diva-146922OAI: oai:DiVA.org:umu-146922DiVA, id: diva2:1203747
Tillgänglig från: 2018-05-04 Skapad: 2018-05-04 Senast uppdaterad: 2019-04-29
Ingår i avhandling
1. On avoiding and completing edge colorings
Öppna denna publikation i ny flik eller fönster >>On avoiding and completing edge colorings
2018 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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. 

Ort, förlag, år, upplaga, sidor
Umeå: Umeå University, 2018. s. 8
Serie
Research report in mathematics, ISSN 1653-0810
Nationell ämneskategori
Diskret matematik
Forskningsämne
matematik
Identifikatorer
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 (Engelska)
Opponent
Handledare
Tillgänglig från: 2018-05-25 Skapad: 2018-05-04 Senast uppdaterad: 2018-06-09Bibliografiskt granskad
2. On avoiding and completing colorings
Öppna denna publikation i ny flik eller fönster >>On avoiding and completing colorings
2019 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

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 Qd, a partial proper edge precoloring φ and a list assignment L must satisfy certain sparsity conditions. Paper II still deals with the 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. 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 K3n,n,n, where n is a power of two in paper IV and n is an even number in paper V.

Ort, förlag, år, upplaga, sidor
Umeå: Umeå University, 2019. s. 11
Serie
Research report in mathematics, ISSN 1653-0810 ; 66/19
Nyckelord
Graph coloring, precoloring, list assignment
Nationell ämneskategori
Diskret matematik
Identifikatorer
urn:nbn:se:umu:diva-158343 (URN)978-91-7855-055-5 (ISBN)
Disputation
2019-05-24, MA121, MIT-huset, Umeå, 10:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2019-05-03 Skapad: 2019-04-24 Senast uppdaterad: 2019-04-29Bibliografiskt granskad

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