319 research outputs found

    Extremal Infinite Graph Theory

    Get PDF
    We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure

    Splitting Line Patterns in Free Groups

    Full text link
    We construct a boundary of a finite rank free group relative to a finite list of conjugacy classes of maximal cyclic subgroups. From the cut points and uncrossed cut pairs of this boundary we construct a simplicial tree on which the group acts cocompactly. We show that the quotient graph of groups is the JSJ decomposition of the group relative to the given collection of conjugacy classes. This provides a characterization of virtually geometric multiwords: they are the multiwords that are built from geometric pieces. In particular, a multiword is virtually geometric if and only if the relative boundary is planar.Comment: 22 pages, 6 figures; v2 fixed a few typos; v3 38 pages, 21 figures; v4 30 pages, 11 figures 'Preliminaries' section expanded to make paper self-contained and split into two sections. Some arguments refactored and simplified. Paper streamlined; v5 56 pages, 21 figures Added examples and improved exposition according to referee comments. To appear in Algebraic & Geometric Topolog

    Combinatorial Properties of Finite Models

    Full text link
    We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite presentation). Extending classical work of Rado (for the random graph), we find a finite presentation for each of the following classes: homogeneous undirected graphs, homogeneous tournaments and homogeneous partially ordered sets. We also give a finite presentation of the rational Urysohn metric space and some homogeneous directed graphs. We survey well known structures that are finitely presented. We focus on structures endowed with natural partial orders and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism orders for various combinatorial objects. We give a new combinatorial proof of the existence of embedding-universal objects for homomorphism-defined classes of structures. This relates countable embedding-universal structures to homomorphism dualities (finite homomorphism-universal structures) and Urysohn metric spaces. Our explicit construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font

    16th Scandinavian Symposium and Workshops on Algorithm Theory: SWAT 2018, June 18-20, 2018, Malmö University, Malmö, Sweden

    Get PDF

    When is a polynomially growing automorphism of FnF_n geometric ?

    Full text link
    The main result of this paper is an algorithmic answer to the question raised in the title, up to replacing the given ϕ^∈Out(Fn)\hat{\phi} \in Out(F_n) by a positive power. In order to provide this algorithm, it is shown that every polynomially growing automorphism ϕ^\hat \phi can be represented by an iterated Dehn twist on some graph-of-groups G\cal{G} with π1G=Fn\pi_1{\cal{G}} = F_n. One then uses results of two previous papers \cite{KY01, KY02} as well as some classical results such as the Whitehead algorithm to prove the claim

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

    Get PDF
    LIPIcs, Volume 248, ISAAC 2022, Complete Volum
    • …
    corecore