12,516 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

    Partitions and Coverings of Trees by Bounded-Degree Subtrees

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    This paper addresses the following questions for a given tree TT and integer dβ‰₯2d\geq2: (1) What is the minimum number of degree-dd subtrees that partition E(T)E(T)? (2) What is the minimum number of degree-dd subtrees that cover E(T)E(T)? We answer the first question by providing an explicit formula for the minimum number of subtrees, and we describe a linear time algorithm that finds the corresponding partition. For the second question, we present a polynomial time algorithm that computes a minimum covering. We then establish a tight bound on the number of subtrees in coverings of trees with given maximum degree and pathwidth. Our results show that pathwidth is the right parameter to consider when studying coverings of trees by degree-3 subtrees. We briefly consider coverings of general graphs by connected subgraphs of bounded degree
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