11,142 research outputs found

    Independent Sets in Graphs with an Excluded Clique Minor

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    Let GG be a graph with nn vertices, with independence number Ξ±\alpha, and with with no Kt+1K_{t+1}-minor for some tβ‰₯5t\geq5. It is proved that (2Ξ±βˆ’1)(2tβˆ’5)β‰₯2nβˆ’5(2\alpha-1)(2t-5)\geq2n-5

    On Tree-Partition-Width

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    A \emph{tree-partition} of a graph GG is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of GG is the minimum number of vertices in a bag in a tree-partition of GG. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width kβ‰₯3k\geq3 and maximum degree Ξ”β‰₯1\Delta\geq1 has tree-partition-width at most 24kΞ”24k\Delta. We prove that this bound is within a constant factor of optimal. In particular, for all kβ‰₯3k\geq3 and for all sufficiently large Ξ”\Delta, we construct a graph with tree-width kk, maximum degree Ξ”\Delta, and tree-partition-width at least (\eighth-\epsilon)k\Delta. Moreover, we slightly improve the upper bound to 5/2(k+1)(7/2Ξ”βˆ’1){5/2}(k+1)({7/2}\Delta-1) without the restriction that kβ‰₯3k\geq3

    Drawing a Graph in a Hypercube

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    A dd-dimensional hypercube drawing of a graph represents the vertices by distinct points in {0,1}d\{0,1\}^d, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.Comment: Submitte

    Colouring the Square of the Cartesian Product of Trees

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    We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree

    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
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