2,070 research outputs found

    A Study of kk-dipath Colourings of Oriented Graphs

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    We examine tt-colourings of oriented graphs in which, for a fixed integer k≥1k \geq 1, vertices joined by a directed path of length at most kk must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case k=2k=2 is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed kk and tt, whether there exists such a tt-colouring are proved.Comment: 14 page

    On First-Order Definable Colorings

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    We address the problem of characterizing HH-coloring problems that are first-order definable on a fixed class of relational structures. In this context, we give several characterizations of a homomorphism dualities arising in a class of structure

    Rips construction without unique product

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    Given a finitely presented group Q,Q, we produce a short exact sequence 1→N↪G↠Q→11\to N \hookrightarrow G \twoheadrightarrow Q \to 1 such that GG is a torsion-free Gromov hyperbolic group without the unique product property and NN is without the unique product property and has Kazhdan's Property (T). Varying Q,Q, we show a wide diversity of concrete examples of Gromov hyperbolic groups without the unique product property. As an application, we obtain Tarski monster groups without the unique product property.Comment: 22 page
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