761 research outputs found

    Defective Coloring on Classes of Perfect Graphs

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    In Defective Coloring we are given a graph GG and two integers χd\chi_d, Δ∗\Delta^* and are asked if we can χd\chi_d-color GG so that the maximum degree induced by any color class is at most Δ∗\Delta^*. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters χd\chi_d, Δ∗\Delta^* is set to the smallest possible fixed value that does not trivialize the problem (χd=2\chi_d = 2 or Δ∗=1\Delta^* = 1). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either χd\chi_d or Δ∗\Delta^* is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both χd\chi_d and Δ∗\Delta^* are unbounded

    Defective and Clustered Graph Colouring

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    Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" dd if each monochromatic component has maximum degree at most dd. A colouring has "clustering" cc if each monochromatic component has at most cc vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding KtK_t as a minor, graphs excluding Ks,tK_{s,t} as a minor, and graphs excluding an arbitrary graph HH as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

    The t-improper chromatic number of random graphs

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    We consider the tt-improper chromatic number of the Erd{\H o}s-R{\'e}nyi random graph G(n,p)G(n,p). The t-improper chromatic number χt(G)\chi^t(G) of GG is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most tt. If t=0t = 0, then this is the usual notion of proper colouring. When the edge probability pp is constant, we provide a detailed description of the asymptotic behaviour of χt(G(n,p))\chi^t(G(n,p)) over the range of choices for the growth of t=t(n)t = t(n).Comment: 12 page

    Ramified rectilinear polygons: coordinatization by dendrons

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    Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic l1l_1-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group D4D_4), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.Comment: 27 pages, 6 figure
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