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Minors in graphs of largeθr-girth

Dimitris Chatzidimitriou, Jean-Florent Raymond, Ignasi Sau, Dimitrios M. Thilikos
2017 European journal of combinatorics (Print)  
Clearly, the girth of a graph is its θ 2 -girth and, for every r 1 ≤ r 2 , the θ r 1 -girth of a graph is at most its θ r 2 -girth.  ...  For every r ∈ N, let θ r denote the graph with two vertices and r parallel edges. The θ r -girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to θ r .  ...  Theorem 3.1 asserts that graphs with large θ r -girth and sufficiently large minimum degree contain as a minor a graph whose minimum degree is exponential in the girth.  ... 
doi:10.1016/j.ejc.2017.04.011 fatcat:eclhqmhbyvhsvcaeyqe2dioxgu

Minors in graphs of large θ_r-girth [article]

Dimitris Chatzidimitriou, Jean-Florent Raymond, Ignasi Sau, Dimitrios M. Thilikos
2017 arXiv   pre-print
In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213--225, 2003], Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is  ...  For every r ∈N, let θ_r denote the graph with two vertices and r parallel edges. The θ_r-girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to θ_r.  ...  As the usual notion of girth appears in various contexts in graph theory, we wonder for which graphs H the results related to girth can be extended to the H-girth or to the two aforementioned variants.  ... 
arXiv:1510.03041v3 fatcat:jqscygckava3nd6rfx2otyyniq

Asymptotics in percolation on high-girth expanders [article]

Michael Krivelevich, Eyal Lubetzky, Benny Sudakov
2020 arXiv   pre-print
We show that, unlike the situation in the classical Erdős-Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size  ...  We consider supercritical bond percolation on a family of high-girth d-regular expanders.  ...  This work was initiated while the authors were visiting the Theory Group of Microsoft Research in Redmond, and they are grateful to the group for its hospitality.  ... 
arXiv:1803.11553v2 fatcat:t5sqzkil75febba5mzcxqdptde

On the Non-Planarity of a Random Subgraph

ALAN FRIEZE, MICHAEL KRIVELEVICH
2013 Combinatorics, probability & computing  
Let G be a finite graph with minimum degree r. Form a random subgraph G p of G by taking each edge of G into G p independently and with probability p.  ...  We prove that for any constant ǫ > 0, if p = 1+ǫ r , then G p is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs.  ...  of girth at least g contains a minor of the complete graph K t .  ... 
doi:10.1017/s0963548313000308 fatcat:qeqrejyopbhtlk5rq2fw3g2cue

Isoperimetric Inequalities and the Width Parameters of Graphs [chapter]

L. Sunil Chandran, T. Kavitha, C. R. Subramanian
2003 Lecture Notes in Computer Science  
The results explained in Section 3 is from Girth and Treewidth (L. S. Chandran, C. R. Subramanian ) [8].  ...  A lower bound for the treewidth in terms of girth and average degree 2.  ...  For a brief exposition of the early developments on the existence of dense minors in graphs of large girth, see Chapter 8 of [11] .  ... 
doi:10.1007/3-540-45071-8_39 fatcat:bf67rueocnhalezo6ewg6jfpuq

Some Recent Progress and Applications in Graph Minor Theory

Ken-ichi Kawarabayashi, Bojan Mohar
2007 Graphs and Combinatorics  
In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the "rough" structure of graphs excluding a fixed minor.  ...  Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed.  ...  Minors in graphs of large girth It was proved by Thomassen [183] that graphs of minimum degree at least three and with large girth contain large clique minors.  ... 
doi:10.1007/s00373-006-0684-x fatcat:wkf3w6cemzc4hnnw3neqpck6hu

On First-Order Definable Colorings [article]

Jaroslav Nesetril
2014 arXiv   pre-print
In this context, we give several characterizations of a homomorphism dualities arising in a class of structure.  ...  We address the problem of characterizing H-coloring problems that are first-order definable on a fixed class of relational structures.  ...  graphs in the class) of p-subdivisions of graphs with arbitrarily large chromatic number and girth.  ... 
arXiv:1403.1995v2 fatcat:httpf736xzfrjmgsfjpyhh7ecq

On first-order definable colorings [chapter]

Jaroslav Nešetřil, Patrice Ossona de Mendez
2014 Geometry, Structure and Randomness in Combinatorics  
In this context, we give several characterizations of a homomorphism dualities arising in a class of structure.  ...  We address the problem of characterizing H-coloring problems that are first-order definable on a fixed class of relational structures.  ...  minors of the graphs in the class.  ... 
doi:10.1007/978-88-7642-525-7_6 fatcat:g2aoxe3jbncwpgevvzzdwcyz6a

Planarity, Colorability, and Minor Games

Dan Hefetz, Michael Krivelevich, Miloš Stojaković, Tibor Szabó
2008 SIAM Journal on Discrete Mathematics  
In this paper we consider the Maker-Breaker and Avoider-Enforcer versions of the planarity game, the k-colorability game and the K t -minor game.  ...  The only difference is in the determination of the winner: Avoider loses if he claims all the vertices of some hyperedge of F; otherwise Enforcer loses.  ...  asserts that in fact it contains a K r -minor.  ... 
doi:10.1137/060654414 fatcat:o55eszjcifaefjjadopsziwrx4

Quantifying Variational Approximation for the Log-Partition Function [article]

Romain Cosson, Devavrat Shah
2021 arXiv   pre-print
In general, κ(G) is the solution of a max-min problem associated with G that can be evaluated in polynomial time for any graph.  ...  As a consequence, the approximation ratio is 1 for trees, √((d+1)/2) for any graph with maximum average degree d, and β→∞≈ 1+1/(2β) for graphs with girth (shortest cycle) at least βlog N.  ...  Acknowledgments This work is supported in parts by projects from NSF and KACST as well as by a Hewlett Packard graduate fellowship.  ... 
arXiv:2102.10196v2 fatcat:r32u5n46wjejzadqwjmhgeohwy

Towards a Theory of Parameterized Streaming Algorithms [article]

Rajesh Chitnis, Graham Cormode
2019 arXiv   pre-print
In this paper, we initiate a systematic study of graph problems from the paradigm of parameterized streaming algorithms.  ...  , Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy.  ...  One of the foundational results of graph theory is the Excluded Grid Minor Theorem of Robertson and Seymour [40] which states that large treewidth forces large grid minors: Theorem 12.  ... 
arXiv:1911.09650v1 fatcat:yvbpw6rsunbcrl3zezonsscf7a

Greedy Learning of Markov Network Structure [article]

Praneeth Netrapalli, Siddhartha Banerjee, Sujay Sanghavi, Sanjay Shakkottai
2012 arXiv   pre-print
Our main result characterizes the sample complexity of this procedure, as a function of node degrees, graph size and girth in factor-graph representation.  ...  For tree graphs, our algorithm is the same as the classical Chow-Liu algorithm, and in that sense can be considered the extension of the same to graphs with cycles.  ...  Johnson for letting us use his graph drawing code for the senator graph.  ... 
arXiv:1202.1787v1 fatcat:j3ulrwktprbjvbj6y7u7xfms3q

On the non-planarity of a random subgraph [article]

Alan Frieze, Michael Krivelevich
2013 arXiv   pre-print
Let G be a finite graph with minimum degree r. Form a random subgraph G_p of G by taking each edge of G into G_p independently and with probability p.  ...  We prove that for any constant ϵ>0, if p=1+ϵ/r, then G_p is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs.  ...  Proof of Theorem 1 Our proof rests in large part on the following simple consequence of Euler's formula. Lemma 1 Let G = (V, E) be a planar graph with n vertices and m edges and girth g.  ... 
arXiv:1205.6240v3 fatcat:y4ulrwmh4zdrxo6tookkneurvu

Optimization of eigenvalue bounds for the independence and chromatic number of graph powers [article]

Aida Abiad, Gabriel Coutinho, Miquel Angel Fiol, Bruno Nogueira, Sjanne Zeijlemaker
2020 arXiv   pre-print
The k^th power of a graph G=(V,E), G^k, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k.  ...  Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.  ...  Acknowledgments The research of A. Abiad is partially supported by the FWO grant 1285921N. A. Abiad and M.A. Fiol gratefully acknowledge the support from DIAMANT. B.  ... 
arXiv:2010.12649v1 fatcat:lhgbg3vhwbgkjc73cox7szpbre

Noncontact Method for Estimating Ellipticity around the Girth of a Free-Ranging Dolphin

Hiromitsu Hama, Tadamichi Morisaka
2021 ICIC Express Letters  
This paper proposes a novel method to estimate the ellipticity around the girth of free-ranging wild dolphins, and clarify their growth and leanness.  ...  In recent years, wild dolphins that live in coastal areas that overlap with areas of human activity are being greatly affected by the increasing conflict with human activities.  ...  Mai Sakai (Kindai University) for providing videos of wild dolphins in Mikura Island, and Ms. Yukie Hashimoto (Otemon Gakuin University) for helping with preliminary experiments in Excel.  ... 
doi:10.24507/icicel.15.07.755 fatcat:gkesozyx5vewvb2mmcxsa5vmbi
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