92,401 research outputs found

    Packing Steiner Trees

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    Let TT be a distinguished subset of vertices in a graph GG. A TT-\emph{Steiner tree} is a subgraph of GG that is a tree and that spans TT. Kriesell conjectured that GG contains kk pairwise edge-disjoint TT-Steiner trees provided that every edge-cut of GG that separates TT has size 2k\ge 2k. When T=V(G)T=V(G) a TT-Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved that Kriesell's conjecture holds when 2k2k is replaced by 24k24k, and recently West and Wu have lowered this value to 6.5k6.5k. Our main result makes a further improvement to 5k+45k+4.Comment: 38 pages, 4 figure

    Generalizations of the Tree Packing Conjecture

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    The Gy\'arf\'as tree packing conjecture asserts that any set of trees with 2,3,...,k2,3, ..., k vertices has an (edge-disjoint) packing into the complete graph on kk vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into kk-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any kk-chromatic graph. We also consider several other generalizations of the conjecture

    The Planar Tree Packing Theorem

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    Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

    Packing and Hausdorff measures of stable trees

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    In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random tree (1991) which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from a joint work with J-F Le Gall (2006).Comment: 40 page
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