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Minors in graphs of largeθr-girth

2017
*
European journal of combinatorics (Print)
*

Clearly, the

doi:10.1016/j.ejc.2017.04.011
fatcat:eclhqmhbyvhsvcaeyqe2dioxgu
*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*. ...##
###
Minors in graphs of large θ_r-girth
[article]

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. ...

##
###
Asymptotics in percolation on high-girth expanders
[article]

2020
*
arXiv
*
pre-print

We show that, unlike the situation

arXiv:1803.11553v2
fatcat:t5sqzkil75febba5mzcxqdptde
*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. ...##
###
On the Non-Planarity of a Random Subgraph

2013
*
Combinatorics, probability & computing
*

Let G be a finite

doi:10.1017/s0963548313000308
fatcat:qeqrejyopbhtlk5rq2fw3g2cue
*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 . ...##
###
Isoperimetric Inequalities and the Width Parameters of Graphs
[chapter]

2003
*
Lecture Notes in Computer Science
*

The results explained

doi:10.1007/3-540-45071-8_39
fatcat:bf67rueocnhalezo6ewg6jfpuq
*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] . ...##
###
Some Recent Progress and Applications in Graph Minor Theory

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*. ...

##
###
On First-Order Definable Colorings
[article]

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*. ...

##
###
On first-order definable colorings
[chapter]

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. ...

##
###
Planarity, Colorability, and Minor Games

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*. ...

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

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. ...

##
###
Towards a Theory of Parameterized Streaming Algorithms
[article]

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. ...

##
###
Greedy Learning of Markov Network Structure
[article]

2012
*
arXiv
*
pre-print

Our main result characterizes the sample complexity

arXiv:1202.1787v1
fatcat:j3ulrwktprbjvbj6y7u7xfms3q
*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*. ...##
###
On the non-planarity of a random subgraph
[article]

2013
*
arXiv
*
pre-print

Let G be a finite

arXiv:1205.6240v3
fatcat:y4ulrwmh4zdrxo6tookkneurvu
*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. ...##
###
Optimization of eigenvalue bounds for the independence and chromatic number of graph powers
[article]

2020
*
arXiv
*
pre-print

The k^th power

arXiv:2010.12649v1
fatcat:lhgbg3vhwbgkjc73cox7szpbre
*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. ...##
###
Noncontact Method for Estimating Ellipticity around the Girth of a Free-Ranging Dolphin

2021
*
ICIC Express Letters
*

This paper proposes a novel method to estimate the ellipticity around the

doi:10.24507/icicel.15.07.755
fatcat:gkesozyx5vewvb2mmcxsa5vmbi
*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. ...
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