14 research outputs found

    Transitive path decompositions of Cartesian products of complete graphs

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    An HH-decomposition of a graph Γ\Gamma is a partition of its edge set into subgraphs isomorphic to HH. A transitive decomposition is a special kind of HH-decomposition that is highly symmetrical in the sense that the subgraphs (copies of HH) are preserved and transitively permuted by a group of automorphisms of Γ\Gamma. This paper concerns transitive HH-decompositions of the graph Kn□KnK_n \Box K_n where HH is a path. When nn is an odd prime, we present a construction for a transitive path decomposition where the paths in the decomposition are arbitrary large.Comment: 14 pages, 4 figure

    The Galois action on Origami curves and a special class of Origamis

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    Origamis are covers of elliptic curves, ramified over at most one point. As they admit a flat atlas, they induce Teichmüller curves in the corresponding moduli spaces of curves. We compare geometric and arithmetic properties of these objects and study in detail a construction given by M. Möller which associates an Origami to a Belyi morphism. This leads to new examples, such as Galois orbits of Origami curves, and an infinite series of non-characteristic Origamis with Veech group SL(2,Z)

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Large bichromatic point sets admit empty monochromatic 4-gons

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    We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum
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