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  • 1.
    Denley, Tristan
    et al.
    Austin Peay State University.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Extending partial latin cubes2014In: Ars combinatoria, ISSN 0381-7032, Vol. 113, p. 405-414Article in journal (Refereed)
    Abstract [en]

    In the spirit of Ryser's theorem, we prove sufficient conditions on k, and m so that k xexm Latin boxes, i.e. partial Latin cubes whose filled cells form a k x x m rectangular box, can be extended to akxnxm latin box, and also to akxnxn latin box, where n is the number of symbols used, and likewise the order of the Latin cube. We also prove a partial Evans type result for Latin cubes, namely that any partial Latin cube of order n with at most n 1 filled cells is completable, given certain conditions on the spatial distribution of the filled cells.

  • 2. Johansson, Anders
    et al.
    Johansson, Robert
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Factors of r-partite graphs and bounds for the strong chromatic number2010In: Ars combinatoria, ISSN 0381-7032, Vol. 95, p. 277-287Article in journal (Refereed)
    Abstract [en]

    We give an optimal degree condition for a tripartite graph to have a spanning subgraph consisting of complete graphs of order 3. This result is used to give an upper bound of 2 Delta for the strong chromatic number of n vertex graphs with Delta >= n/6.

  • 3.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Complete minors in cubic graphs with few short cycles and random cubic graphs2004In: Ars combinatoria, ISSN 0381-7032, Vol. 70, p. 289-295Article in journal (Refereed)
    Abstract [en]

    We first prove that for any fixed kappa a cubic graph with few short cycles contains a K-kappa-minor. This is a direct generalisation of a result on girth by Thomassen. We then use this theorem to show that for any fixed k a random cubic graph contains a Kk-minor asymptotically almost surely.

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