311 research outputs found

    A survey of degree-boundedness

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    If a graph has no large balanced bicliques, but has large minimum degree, then what can we say about its induced subgraphs? This question motivates the study of degree-boundedness, which is like χ\chi-boundedness but for minimum degree instead of chromatic number. We survey this area with an eye towards open problems

    Geometric Graphs with Unbounded Flip-Width

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    We consider the flip-width of geometric graphs, a notion of graph width recently introduced by Toru\'nczyk. We prove that many different types of geometric graphs have unbounded flip-width. These include interval graphs, permutation graphs, circle graphs, intersection graphs of axis-aligned line segments or axis-aligned unit squares, unit distance graphs, unit disk graphs, visibility graphs of simple polygons, β\beta-skeletons, 4-polytopes, rectangle of influence graphs, and 3d Delaunay triangulations.Comment: 10 pages, 7 figures. To appear at CCCG 202

    Prime and polynomial distances in colourings of the plane

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    We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial ff with integer coefficients and positive leading coefficient, every finite colouring of the plane contains a monochromatic pair of distinct points whose distance is equal to f(n)f(n) for some integer nn. The second is that for every finite colouring of the plane, there is a monochromatic pair of points whose distance is a prime number.Comment: 22 page

    Local Structure for Vertex-Minors

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    This thesis is about a conjecture of Geelen on the structure of graphs with a forbidden vertex-minor; the conjecture is like the Graph Minors Structure Theorem of Robertson and Seymour but for vertex-minors instead of minors. We take a step towards proving the conjecture by determining the "local structure''. Our first main theorem is a grid theorem for vertex-minors, and our second main theorem is more like the Flat Wall Theorem of Robertson and Seymour. We believe that the results presented in this thesis provide a path towards proving the full conjecture. To make this area more accessible, we have organized the first chapter as a survey on "structure for vertex-minors''

    Colouring Polygon Visibility Graphs and Their Generalizations

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    Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time
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