52 research outputs found

    Hamiltonicity problems in random graphs

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    In this thesis, we present some of the main results proved by the author while fulfilling his PhD. While we present all the relevant results in the introduction of the thesis, we have chosen to focus on two of the main ones. First, we show a very recent development about Hamiltonicity in random subgraphs of the hypercube, where we have resolved a long standing conjecture dating back to the 1980s. Second, we present some original results about correlations between the appearance of edges in random regular hypergraphs, which have many applications in the study of subgraphs of random regular hypergraphs. In particular, these applications include subgraph counts and property testing

    Packing and embedding large subgraphs

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    This thesis contains several embedding results for graphs in both random and non random settings. Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals 1/21/2. %posed e.g.~by Bollob\'as, In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n: if H\subseteq\cQ^n satisfies ÎŽ(H)≄αn\delta(H)\geq\alpha n with α>0\alpha>0 fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with p∈(0,1]p\in(0,1] fixed, then with high probability H\cup\cQ^n_p contains kk edge-disjoint Hamilton cycles, for any fixed k∈Nk\in\mathbb{N}. This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity. In Chapter 3 we move to a non random setting. %to a deterministic one. %Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph. Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs. More specifically, we provide a degree condition on a regular nn-vertex graph GG which ensures the existence of a near optimal packing of any family H\mathcal H of bounded degree nn-vertex kk-chromatic separable graphs into GG. %In general, this degree condition is best possible. %In particular, this yields an approximate version of the tree packing conjecture %in the setting of regular host graphs GG of high degree. %Similarly, our result implies approximate versions of the Oberwolfach problem, %the Alspach problem and the existence of resolvable designs in the setting of %regular host graphs of high degree. In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Modelling Distribution Routes in City Logistics by Applying Operations Research Methods

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    The article focuses on the up-to-date subject from the practical as well as scientific point of view. It specifically discusses a proposal of an approach concerning transport or distribution problems in the range of city logistics and investigates possibilities to use opted operations research methods in this particular area. Specific suggestions lie first and foremost in using selected tools of operations research (i.e. a set of methods concerning vehicle routing problem) to model multiple variants of distribution paths from a determined hub to multiple spokes in order to minimise the overall travelled distance in an urban area. As far as the very research goes, to define distribution paths to supply multiple logistics objects in the range of city logistics, ensuing methods are step by step used: Clarke-Wright algorithm, Mayer algorithm and the nearest neighbour algorithm. The article consists of a conceptual section, describing the relevant theory as well as data and methods used, the practical part and the section encompassing an assessment of the key findings, along with the discussion. A suitable combination of adequate operations research methods and their application to city logistics issues is where an innovative solution of this research lies

    27th Annual European Symposium on Algorithms: ESA 2019, September 9-11, 2019, Munich/Garching, Germany

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    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Genetic neural networks on MIMD computers

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    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Proceedings of the 10th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications

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