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

Raphael Yuster
2010 arXiv   pre-print
It is proved that 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.  ... 
arXiv:1009.0353v1 fatcat:tqpqaj4vonbmzi5bkw6h4ee7bu

Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing

RAPHAEL YUSTER
2012 Combinatorics, probability & computing  
We prove that 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.  ... 
doi:10.1017/s0963548312000235 fatcat:lgmrtfjk3jczbcpbwdocye73kq

On the Maximum Quadratic Assignment Problem [chapter]

Viswanath Nagarajan, Maxim Sviridenko
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
To find these subgraphs, we use a 2-approximation algorithm for vertex 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.  ... 
doi:10.1137/1.9781611973068.57 fatcat:ppbi2ajpprgn7jppu3dz6ex6j4

On the Maximum Quadratic Assignment Problem

Viswanath Nagarajan, Maxim Sviridenko
2009 Mathematics of Operations Research  
To find these subgraphs, we use a 2-approximation algorithm for vertex 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.  ... 
doi:10.1287/moor.1090.0418 fatcat:g5xg6p47gjbvlkjnjku4nrl3z4

Hamiltonian cycles on a random three-coordinate lattice

B. Eynard, E. Guitter, C. Kristjansen
1998 Nuclear Physics B  
Consider a 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 .  ... 
doi:10.1016/s0550-3213(98)00391-5 fatcat:mhse7foslvc6xhbwtx4wptvaly

An approximate method for circle packing and disc covering

Shuangmin Chen, Yuen-Shan Leung, Shiqing Xin, Ying He, Yuanfeng Zhou, Changhe Tu
2018 Communications in Information and Systems  
Observing that circle 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.  ... 
doi:10.4310/cis.2018.v18.n2.a1 fatcat:ratgru6f4zdhljyhithr7ffxey

Experimental Study of Independent and Dominating Sets in Wireless Sensor Networks Using Graph Coloring Algorithms [chapter]

Dhia Mahjoub, David W. Matula
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.  ... 
doi:10.1007/978-3-642-03417-6_4 fatcat:hbvpocsncbcwhmic75suu76asq

Dense Packing on Uniform Lattices

Kari Eloranta
2007 Journal of statistical physics  
Our particular aim 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.  ... 
doi:10.1007/s10955-007-9448-0 fatcat:f44mzeiqwnbkfoqfvzojtol7dm

Periodic Planar Disk Packings [article]

Robert Connelly, William Dickinson
2013 arXiv   pre-print
Conjectures are presented that claim that the density of any strictly jammed 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.  ... 
arXiv:1201.5965v2 fatcat:pif63nopancqxi6ybo2xoqzqaq

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  ... 

Packing edge-disjoint triangles in regular and almost regular tournaments

Islam Akaria, Raphael Yuster
2015 Discrete Mathematics  
For a tournament T , let ν 3 (T ) denote the maximum number of pairwise edge-disjoint 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 ≡  ... 
doi:10.1016/j.disc.2014.09.010 fatcat:4qwwdvmqz5hqrcfkz4hyemjojy

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

David J. Aldous, Colin McDiarmid, Alex Scott
2009 Operations Research Letters  
The proof uses a reduction to a 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  ... 
doi:10.1016/j.orl.2009.04.008 fatcat:tywh3s6kbneblixf2omex5achy

Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles [article]

Xujin Chen, Zhuo Diao, Xiaodong Hu, Zhongzheng Tang
2016 arXiv   pre-print
These algorithms imply new sufficient conditions for Tuza's conjecture on 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.  ... 
arXiv:1605.01816v3 fatcat:ntstoa3windgpcyw5szaxn4jve

Periodic planar disc packings

R. Connelly, W. Dickinson
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.  ... 
doi:10.1098/rsta.2012.0039 pmid:24379429 fatcat:3gjt56o6ing7hhaj4k5jazpupi
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