217 research outputs found

    On the oriented chromatic number of dense graphs

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    Let GG be a graph with nn vertices, mm edges, average degree δ\delta, and maximum degree Δ\Delta. The \emph{oriented chromatic number} of GG is the maximum, taken over all orientations of GG, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which δlogn\delta\geq\log n. We prove that every such graph has oriented chromatic number at least Ω(n)\Omega(\sqrt{n}). In the case that δ(2+ϵ)logn\delta\geq(2+\epsilon)\log n, this lower bound is improved to Ω(m)\Omega(\sqrt{m}). Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when GG is (clognc\log n)-regular for some constant c>2c>2, in which case the oriented chromatic number is between Ω(nlogn)\Omega(\sqrt{n\log n}) and O(nlogn)\mathcal{O}(\sqrt{n}\log n)

    Complete Acyclic Colorings

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    We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure

    Upward Three-Dimensional Grid Drawings of Graphs

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    A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every nn-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O(n3/2)O(n^{3/2}) volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with (n3)(n^3) volume, which is tight for the complete dag. The previous best upper bound was O(n4)O(n^4). Our main result is that every cc-colourable directed acyclic graph (cc constant) has an upward three-dimensional grid drawing with O(n2)O(n^2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n3)O(n^3) for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar

    Asymmetric coloring games on incomparability graphs

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    Consider the following game on a graph GG: Alice and Bob take turns coloring the vertices of GG properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of GG is the minimum number of colors that allows Alice to win the game. The game Grundy number of GG is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The (a,b)(a,b)-game chromatic and Grundy numbers are defined likewise except that Alice colors aa vertices and Bob colors bb vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs (a,b)(a,b) for which the (a,b)(a,b)-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu
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