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Small minors in dense graphs

Samuel Fiorini, Gwenaël Joret, Dirk Oliver Theis, David R. Wood
2012 European journal of combinatorics (Print)  
We prove this result with the extra property that the minor is small with respect to the order of the whole graph.  ...  A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor.  ...  Thanks to Michele Conforti for suggesting to study the relationship between average degree and small models. Thanks to Paul Seymour for suggesting the example following Lemma 4.2.  ... 
doi:10.1016/j.ejc.2012.02.003 fatcat:5e4ixq3kx5e4xldas4npgic4ve

Sparse Combinatorial Structures: Classification and Applications

Jaroslav Nešetřil, Patrice Ossona de Mendez
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
All these problems can be studied under the umbrella of classes of structures which are Nowhere Dense and in the context of Nowhere Dense -Somewhere Dense dichotomy.  ...  We present results of the recent research on sparse graphs and finite structures in the context of of contemporary combinatorics, graph theory, model theory and mathematical logic, complexity of algorithms  ...  Thus C is somewhere dense if every graph is a bounded depth shallow minor of a graph in C. In other words: we get all graphs in a fixed time.  ... 
doi:10.1142/9789814324359_0156 fatcat:io4dnuj4wzarvest64wnvvm4ge

Constructing dense graphs with sublinear Hadwiger number [article]

Jacob Fox
2011 arXiv   pre-print
Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices.  ...  We answer these questions by showing how to explicitly construct such graphs using blow-ups of small graphs with this property.  ...  Thus, if we found a dense graph G with relatively small fractional Hadwiger number, then the blow-up G[r] would also be dense and have relatively small Hadwiger number.  ... 
arXiv:1108.4953v2 fatcat:sgcjyvu5afe33eodi5bfvnz77i

Fixed-parameter Tractable Distances to Sparse Graph Classes [article]

Jannis Bulian, Anuj Dawar
2015 arXiv   pre-print
The second shows that determining the elimination distance of a graph G to a minor-closed class C is FPT.  ...  The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT.  ...  In nowhere dense classes of graphs, we cannot in general bound the size of neighbourhoods.  ... 
arXiv:1502.05910v1 fatcat:uhxviaio4nfo3aahfvdks3eeuq

Linear-sized minors with given edge density [article]

Tung H. Nguyen
2022 arXiv   pre-print
with no K_t minor.  ...  It is proved that for every ε>0, there exists K>0 such that for every integer t≥2, every graph with chromatic number at least Kt contains a minor with t vertices and edge density at least 1-ε.  ...  This work was partially done at the workshop "Seymour is 70 + 2ε" at LIP, ENS de Lyon in June, 2022.  ... 
arXiv:2206.14309v2 fatcat:lynnaal2irgodiwdyorn4wxi4m

Local Structure Theorems for Erdos Renyi Graphs and their Algorithmic Application [article]

Jan Dreier, Philipp Kuinke, Ba Le Xuan, Peter Rossmanith
2018 arXiv   pre-print
This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter.  ...  We analyze some local properties of sparse Erdos-Renyi graphs, where d(n)/n is the edge probability. In particular we study the behavior of very short paths.  ...  As the dense part is quite small it gives us hope that hybrid algorithms exist that combine different methods for the dense part and the structurally simple part.  ... 
arXiv:1709.09152v2 fatcat:zgpfy23yvbdcngrsavnaomgjqa

Local Structure Theorems for Erdős–Rényi Graphs and Their Algorithmic Applications [chapter]

Jan Dreier, Philipp Kuinke, Ba Le Xuan, Peter Rossmanith
2017 Lecture Notes in Computer Science  
This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter.  ...  We analyze local properties of sparse Erdős-Rényi graphs, where d(n)/n is the edge probability. In particular we study the behavior of very short paths.  ...  As the dense part is quite small it gives us hope that hybrid algorithms exist that combine different methods for the dense part and the structurally simple part.  ... 
doi:10.1007/978-3-319-73117-9_9 fatcat:bmersafop5axzio3e5zxjdgdbu

Towards a Characterization of Universal Categories [article]

J. Nesetril, P. Ossona de Mendez
2016 arXiv   pre-print
In this note we characterize, within the framework of the theory of finite set, those categories of graphs that are algebraic universal in the sense that every concrete category embeds in them.  ...  The proof of the characterization is based on the sparse--dense dichotomy and its model theoretic equivalent.  ...  expansion proper topological minor closed Concrete categories somewhere dense nowhere dense 3.  ... 
arXiv:1608.01112v1 fatcat:2rq6yp3aizfm7lc6kizxsus7hm

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-Wideness [article]

Wojciech Nadara and Marcin Pilipczuk and Roman Rabinovich and Felix Reidl and Sebastian Siebertz
2019 arXiv   pre-print
of this algorithm are close to optimal in graph classes with fixed excluded minor.  ...  The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important  ...  Furthermore, we thank anonymous reviewers for their very valuable and in-depth comments on the manuscript.  ... 
arXiv:1802.09801v2 fatcat:uzt2w6m2z5hj7fp6ca6glkusl4

Structural Properties of Sparse Graphs

Jaroslav Nešetřil
2008 Electronic Notes in Discrete Mathematics  
Dense Graphs Dense graphs have been extensively studied in the context of Extremal Graph Theory.  ...  [100] introduced the concept of limited-depth minor exclusion and have shown that exclusion of small limited-depth minors implies the existence of a small separator.  ... 
doi:10.1016/j.endm.2008.06.050 fatcat:hk5istkcqzftjnfm7kqgsia7sm

Structural Properties of Sparse Graphs [chapter]

Jaroslav NeŠetřil, Patrice Ossona De Mendez
2008 Bolyai Society Mathematical Studies  
Dense Graphs Dense graphs have been extensively studied in the context of Extremal Graph Theory.  ...  [100] introduced the concept of limited-depth minor exclusion and have shown that exclusion of small limited-depth minors implies the existence of a small separator.  ... 
doi:10.1007/978-3-540-85221-6_13 fatcat:j7ifp6snovdybopviit5zaya2e

Breaking the degeneracy barrier for coloring graphs with no K_t minor [article]

Sergey Norin, Luke Postle, Zi-Xia Song
2020 arXiv   pre-print
In 1943, Hadwiger conjectured that every graph with no K_t minor is (t-1)-colorable for every t≥ 1.  ...  In the 1980s, Kostochka and Thomason independently proved that every graph with no K_t minor has average degree O(t√(log t)) and hence is O(t√(log t))-colorable.  ...  In [NS19a] the first and third author have shown that every graph with no K t minor is O(t(log t) 0.354 )-colorable.  ... 
arXiv:1910.09378v2 fatcat:ye22iq46njhlrglp56woiaoffu

The Extremal Function for Complete Minors

Andrew Thomason
2001 Journal of combinatorial theory. Series B (Print)  
EXTREMAL FUNCTION FOR COMPLETE MINORS  ...  This is explained in Section 2.  ...  But nothing was proved about dense graphs of very small order, where random graphs are expected to provide the extremal examples.  ... 
doi:10.1006/jctb.2000.2013 fatcat:xwnf7y2im5auhcmpdgwz6slrd4

On nowhere dense graphs

Jaroslav Nešetřil, Patrice Ossona de Mendez
2011 European journal of combinatorics (Print)  
In this paper, we define and analyze the nowhere dense classes of graphs.  ...  In this paper, we show that this concept leads to a classification of general classes of graphs and to the dichotomy between nowhere dense and somewhere dense classes.  ...  Moreover, we will demand that our definition stays invariant in the context of derived classes, i.e. when we perform lexicographic products with small graphs, contractions of small balls, etc.  ... 
doi:10.1016/j.ejc.2011.01.006 fatcat:m4xrtlurfrev7divuibohk4fsu

Reducing Linear Hadwiger's Conjecture to Coloring Small Graphs [article]

Michelle Delcourt, Luke Postle
2022 arXiv   pre-print
In 1943, Hadwiger conjectured that every graph with no K_t minor is (t-1)-colorable for every t≥ 1.  ...  First as mentioned, using the current best-known bounds on coloring small K_t-minor-free graphs, we show that K_t-minor-free graphs are O(tloglog t)-colorable.  ...  Acknowledgments We thank Paul Seymour for bringing to our attention that Lemma 5.14 was already known in the literature, namely in Kawarabayashi's paper from 2007 [11] .  ... 
arXiv:2108.01633v3 fatcat:upkl4bszlzbmfltopp7kinmqdq
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