691 research outputs found

    Edge-disjoint Hamilton cycles in graphs

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    In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least (1/2+α)n(1/2 + \alpha)n can be almost decomposed into edge-disjoint Hamilton cycles.Comment: Minor Revisio

    Schnyder decompositions for regular plane graphs and application to drawing

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    Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to dd-angulations (plane graphs with faces of degree dd) for all d≥3d\geq 3. A \emph{Schnyder decomposition} is a set of dd spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d−2d-2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the dd-angulation is dd. As in the case of Schnyder woods (d=3d=3), there are alternative formulations in terms of orientations ("fractional" orientations when d≥5d\geq 5) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed dd-angulation of girth dd is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on dd-regular plane graphs of mincut dd rooted at a vertex v∗v^*) are decompositions into dd spanning trees rooted at v∗v^* such that each edge not incident to v∗v^* is used in opposite directions by two trees. Additionally, for even values of dd, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder decompositions yields (planar) orthogonal and straight-line drawing algorithms. For a 4-regular plane graph GG of mincut 4 with nn vertices plus a marked vertex vv, the vertices of G\vG\backslash v are placed on a (n−1)×(n−1)(n-1) \times (n-1) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n−22n-2 edges of G\vG\backslash v has exactly one bend. Embedding also the marked vertex vv is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to vv. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around 25n/32×25n/3225n/32\times 25n/32 for a uniformly random instance with nn vertices

    Processes on Unimodular Random Networks

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    We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version is incorrect --, as well as a minor error in the proof of Proposition 4.10; 4th version corrects proof of Proposition 7.1; 5th version corrects proof of Theorem 5.1; 6th version makes a few more minor correction
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