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Dense graphs with a large triangle cover have a large triangle packing
[article]

2010
*
arXiv
*
pre-print

It is proved that

arXiv:1009.0353v1
fatcat:tqpqaj4vonbmzi5bkw6h4ee7bu
*dense**graphs*that are hard to make*triangle*-free, have a large*packing*of pairwise edge-disjoint*triangles*. ...*In*particular, they have more than m(1/4+cβ^2) pairwise edge-disjoint*triangles*where β is the density of the*graph**and*c is an absolute constant. ... Acknowledgment I thank Guy Kortsarz*and*Michael Krivelevich for useful discussions. ...##
###
Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing

2012
*
Combinatorics, probability & computing
*

We prove that

doi:10.1017/s0963548312000235
fatcat:lgmrtfjk3jczbcpbwdocye73kq
*dense**graphs*that are hard to make*triangle*-free have a large*packing*of pairwise edge-disjoint*triangles*. ... We conjecture that such*graphs*have an asymptotically optimal*triangle**packing*of size m(1/3 − o(1)). We extend our result from*triangles*to larger cliques*and*odd cycles. ... Acknowledgment I thank Guy Kortsarz*and*Michael Krivelevich for useful discussions*and*the referee for valuable comments. ...##
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On the Maximum Quadratic Assignment Problem
[chapter]

2009
*
Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
*

To find these subgraphs, we use a 2-approximation algorithm for vertex

doi:10.1137/1.9781611973068.57
fatcat:ppbi2ajpprgn7jppu3dz6ex6j4
*cover**in*one*graph*,*and*an n/k-approximation algorithm for*dense*k subgraph*in*the other*graph*; (2) The second algorithm uses local ... Quadratic assignment is a basic problem*in*combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement,*dense*k subgraph,*and*clustering with given ... Given two undirected*graphs*G*and*H , a common star*packing*consists of star*packings**in*G*and**in*H such that*and*have identical size vectors. ...##
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On the Maximum Quadratic Assignment Problem

2009
*
Mathematics of Operations Research
*

To find these subgraphs, we use a 2-approximation algorithm for vertex

doi:10.1287/moor.1090.0418
fatcat:g5xg6p47gjbvlkjnjku4nrl3z4
*cover**in*one*graph*,*and*an n/k-approximation algorithm for*dense*k subgraph*in*the other*graph*; (2) The second algorithm uses local ... Quadratic assignment is a basic problem*in*combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement,*dense*k subgraph,*and*clustering with given ... Given two undirected*graphs*G*and*H , a common star*packing*consists of star*packings**in*G*and**in*H such that*and*have identical size vectors. ...##
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Hamiltonian cycles on a random three-coordinate lattice

1998
*
Nuclear Physics B
*

Consider a

doi:10.1016/s0550-3213(98)00391-5
fatcat:mhse7foslvc6xhbwtx4wptvaly
*random*three-coordinate lattice of spherical topology having 2v vertices*and*being*densely**covered*by a single closed, self-avoiding walk, i.e. being equipped with a Hamiltonian cycle. ... Eynard acknowledges the support*and*the hospitality of the Niels Bohr Institute as well as the support of the European Union through their TMR programme, contract No. ERBFMRXCT 960012. We thank F. ...*and**densely**covered*by i closed, self-avoiding*and*non-intersecting loops 4 . ...##
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An approximate method for circle packing and disc covering

2018
*
Communications in Information and Systems
*

Observing that circle

doi:10.4310/cis.2018.v18.n2.a1
fatcat:ratgru6f4zdhljyhithr7ffxey
*packing*is closely related to disc*covering*, we formulate both problems*in*a centroidal Voronoi tessellation (CVT)-like computational framework*and*show that a locally optimal solution ...*In*this paper, we develop an approximate method for*packing*a large number of circles (of similar sizes). ... Acknowledgement We are very grateful to the editors*and*anonymous reviewers for their insightful comments*and*suggestions. ...##
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Experimental Study of Independent and Dominating Sets in Wireless Sensor Networks Using Graph Coloring Algorithms
[chapter]

2009
*
Lecture Notes in Computer Science
*

*In*this study, we address the domatic partition problem

*in*

*random*geometric

*graphs*by investigating several vertex coloring algorithms both topology

*and*geometry-aware, color-adaptive

*and*

*randomized*. ... We also observe that

*in*

*dense*

*random*deployments, certain coloring algorithms yield a

*packing*of the nodes into independent sets each of which is relatively close to the perfect placement

*in*the triangular ... Mihaela Iridon

*and*Dr. Saeed Abu-Nimeh for several valuable discussions

*and*Ilteris Murat Derici for his initial work on the SL algorithm

*and*RGG. ...

##
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Dense Packing on Uniform Lattices

2007
*
Journal of statistical physics
*

Our particular aim

doi:10.1007/s10955-007-9448-0
fatcat:f44mzeiqwnbkfoqfvzojtol7dm
*in*choosing these*graphs*is to obtain insight to the geometry of the densest*packings**in*a uniform discrete set-up. ... These phenomena are related to the exponential size of the set of densest*packings**and*more specifically whether these*packings*are maximally symmetric, simple laminated or essentially*random**packings*. ... Acknowledgements The author would like to thank Joel Lebowitz for discussions on these models*and*the referees for criticism that led to improvement*and**in*particular to the completion of Table I. ...##
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Periodic Planar Disk Packings
[article]

2013
*
arXiv
*
pre-print

Conjectures are presented that claim that the density of any strictly jammed

arXiv:1201.5965v2
fatcat:pif63nopancqxi6ybo2xoqzqaq
*packings*, whose*graph*does not consist of all*triangles**and*the torus lattice is the standard triangular lattice, is at most ... Several conditions are given when a*packing*of equal disks*in*a torus is locally maximally*dense*, where the torus is defined as the quotient of the plane by a two-dimensional lattice. ...*In*[14]*and*[13] they estimated the ratio of squares to*triangles**in*a "*random*" square-*and*-*triangle*tiling. Square lattice tilings were quite popular. ...##
###
Page 3580 of Mathematical Reviews Vol. , Issue 2004e
[page]

2004
*
Mathematical Reviews
*

Isaak (1-LEHI; Bethlehem, PA) 2004e:05153 05C70 05B40 Liang, Zhihe (PRC-HNU; Shijiazhuang)

*Graph*designs,*packings**and**coverings*of 1K, with a*graph*of six vertices*and*containing a*triangle*. ... From the summary: “*In*this paper, we determine the existence COMBINATORICS 3580 spectrum for the G-designs of 4K,, 42> 1,*and*construct the maximum*packings**and*the minimum*coverings*of AK, with G for ...##
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Packing edge-disjoint triangles in regular and almost regular tournaments

2015
*
Discrete Mathematics
*

For a tournament T , let ν 3 (T ) denote the maximum number of pairwise edge-disjoint

doi:10.1016/j.disc.2014.09.010
fatcat:4qwwdvmqz5hqrcfkz4hyemjojy
*triangles*(directed cycles of length 3)*in*T . ... We conjecture that ν 3 (n) = (1 + o(1))n 2 /9*and*prove that n 2 11.43 improving upon the best known upper bound of n 2 −1 8*and*lower bound of n 2 11.5 (1 − o(1)). ... The study of*triangle**packings**in**graphs*was initiated*in*the classical result of Kirkman [8] who proved that the complete*graph*with n vertices has a*triangle**packing*of size n(n − 1)/6 whenever n ≡ ...##
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Page 1495 of Mathematical Reviews Vol. , Issue 2001C
[page]

2001
*
Mathematical Reviews
*

Balakrishnan (6-TRCH; Tiruchirapalli)
2001¢:05120 05C75
Brandt, Stephan (D-FUB-C; Berlin)
On the structure of

*dense**triangle*-free*graphs*. (English summary) Combin. Probab. ...*In*particular, results of G. Z. Liu*and*H. Feng [“2-factors with some properties*in*2d-regular*graphs*”,*in*Combinatorics*and**graph*theory ‘97, Vol. 1, World Sci. ...##
###
Uniform multicommodity flow through the complete graph with random edge-capacities

2009
*
Operations Research Letters
*

The proof uses a reduction to a

doi:10.1016/j.orl.2009.04.008
fatcat:tywh3s6kbneblixf2omex5achy
*random**graph*problem. ... We prove that Φ n → φ*in*probability where the limit constant φ depends on the distribution of C*in*a simple way,*and*that asymptotically one need use only oneand two-step routes. ... We now use Theorem 1 of Haxell*and*Rödl [9] , which implies that the maximum size of a*triangle*-*packing**in*a*graph*on m vertices differs from the maximum size of a fractional*triangle*-*packing*by o(m 2 ...##
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Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
[article]

2016
*
arXiv
*
pre-print

These algorithms imply new sufficient conditions for Tuza's conjecture on

arXiv:1605.01816v3
fatcat:ntstoa3windgpcyw5szaxn4jve
*covering**and**packing**triangles*. More precisely, suppose that the set $\mathscr T_G$ of*triangles**covers*all edges*in*$G$. ... Let $\nu_t(G)$*and*$\tau_t(G)$ denote the maximum number of pairwise edge-disjoint*triangles**in*$G$*and*the minimum cardinality of a*triangle**cover*of $G$, respectively. ... Puleo*and*Zbigniew Lonc for their invaluable comments*and*suggestions which have greatly improved the presentation of this paper. ...##
###
Periodic planar disc packings

2013
*
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
*

*In*[19]

*and*[18] , the authors estimated the ratio of squares to

*triangles*

*in*a '

*random*' square-

*and*-

*triangle*tiling. Square lattice tilings were quite popular. ... The whole

*graph*is a

*covering*

*graph*of a four-vertex

*graph*on another lattice Λ that is not a triangular lattice. ...

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