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Petersen-colorings and some families of snarks
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Paderborn institute for advanced studies in Computer science and Engineering, Paderborn University.
2014 (English)In: Ars Mathematica Contemporanea, ISSN 1855-3966, Vol. 7, no 1, 161-173 p.Article in journal (Other academic) Published
Abstract [en]

In this paper we study Petersen-colorings and strong Petersen-colorings on some well known families of snarks, e.g. Blanusa snarks, Goldberg snarks and flower snarks. In particular, it is shown that flower snarks have a Petersen-coloring but they do not have a strong Petersen-coloring. Furthermore it is proved that possible minimum counterexamples to Jaeger's Petersen-coloring conjecture do not contain a specific subdivision of K-3,K-3.

Place, publisher, year, edition, pages
2014. Vol. 7, no 1, 161-173 p.
Keyword [en]
Petersen colorings, snarks
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-53324ISI: 000320236500013OAI: oai:DiVA.org:umu-53324DiVA: diva2:511375
Available from: 2012-03-21 Created: 2012-03-21 Last updated: 2017-12-07Bibliographically approved
In thesis
1. Snarks: Generation, coverings and colourings
Open this publication in new window or tab >>Snarks: Generation, coverings and colourings
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

For a number of unsolved problems in graph theory such as the cycle double cover conjecture, Fulkerson's conjecture and Tutte's 5-flow conjecture it is sufficient to prove them for a family of graphs called snarks. Named after the mysterious creature in Lewis Carroll's poem, a \emph{snark} is a cyclically 4-edge connected 3-regular graph of girth at least 5 which cannot be properly edge coloured using three colours. Snarks and problems for which an edge minimal counterexample must be a snark are the central topics of this thesis.  

The first part of this thesis is intended as a short introduction to the area. The second part is an introduction to the appended papers and the third part consists of the four papers presented in a chronological order.

In Paper I we study the strong cycle double cover conjecture and stable cycles for small snarks. We prove that if a bridgeless cubic graph $G$ has a cycle of length at least $|V(G)|-9$ then it also has a cycle double cover. Furthermore we show that there exist cyclically 5-edge connected snarks with stable cycles and that there exists an infinite family of snarks with stable cycles of length 24.

In Paper II we present a new algorithm for generating all non-isomorphic snarks with a given number of vertices. We generate all snarks on 36 vertices and less and study these with respect to various properties. We find that a number of conjectures on cycle covers and colourings holds for all graphs of these orders. Furthermore we present counterexamples to no less than eight published conjectures on cycle coverings, cycle decompositions and the general structure of regular graphs.    

In Paper III we show that Jaeger's Petersen colouring conjecture holds for three infinite families of snarks and that a minimum counterexample to this conjecture cannot contain a certain subdivision of $K_{3,3}$ as a subgraph. Furthermore, it is shown that one infinite family of snarks have strong Petersen colourings while another does not have any such colourings.

Two simple constructions for snarks with arbitrary high oddness and resistance is given in Paper IV. It is observed that some snarks obtained from this construction have the property that they require at least five perfect matchings to cover the edges. This disproves a suggested strengthening of Fulkerson's conjecture.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2012. 69 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 53
Keyword
Snarks, cubic graphs, enumeration, cycle covers, perfect matching covers, Fulkerson's conjecture, strong cycle double cover conjecture, cycle decompositions, stable cycles, oddness, circumference, uncolourability measures
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-53337 (URN)978-91-7459-399-0 (ISBN)
Public defence
2012-04-13, MIT-huset, MA121, Umeå universitet, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2012-03-23 Created: 2012-03-21 Last updated: 2013-09-05Bibliographically approved

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Hägglund, Jonas

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