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Bounds on the chromatic number of intersection graphs of sets in the plane

I.G. Perepelitsa
2003 Discrete Mathematics  
In this paper we show that the chromatic number of intersection graphs of congruent geometric ÿgures obtained by translations of a ÿxed ÿgure in the plane is bounded by the clique number.  ...  Further, in the paper we prove that the triangle-free intersection graph of a ÿnite number of compact connected sets with piecewise di erentiable Jordan curve boundaries is planar and hence, is 3-colorable  ...  Introduction An interesting problem of estimating the maximum chromatic number of intersection graphs on the plane was started by Asplund and Grűnbaum [2] .  ... 
doi:10.1016/s0012-365x(02)00501-0 fatcat:46e5neusdrabvdbqmwescupt6a

Triangle-free intersection graphs of line segments with large chromatic number

Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, Bartosz Walczak
2014 Journal of combinatorial theory. Series B (Print)  
In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no.  ...  Specifically, for each positive integer k, we construct a triangle-free family of line segments in the plane with chromatic number greater than k.  ...  The intersection graph of a family of sets F is the graph with vertex set F and edge set consisting of pairs of intersecting elements of F.  ... 
doi:10.1016/j.jctb.2013.11.001 fatcat:nrsq24zcurbjflw5ojs2j3hcgy

Box and segment intersection graphs with large girth and chromatic number [article]

James Davies
2021 arXiv   pre-print
We prove that there are intersection graphs of axis-aligned boxes in ℝ^3 and of straight line segments in ℝ^3 that have arbitrarily large girth and chromatic number.  ...  The author would also like to thank Tom Johnston for pointing out an issue in the proof of Theorem 6 in a previous version of this manuscript.  ...  Acknowledgements The author would like to thank Jim Geelen, Matthew Kroeker, and Rose McCarty for helpful discussions, which in particular improved the proof of Theorem 1.  ... 
arXiv:2011.14174v2 fatcat:hnu3542pu5a5lkefd7zyqzljpm

Note on the number of edges in families with linear union-complexity [article]

Piotr Micek, Rom Pinchasi
2014 arXiv   pre-print
We give a simple argument showing that the number of edges in the intersection graph G of a family of n sets in the plane with a linear union-complexity is O(ω(G)n).  ...  In particular, we prove χ(G)≤col(G)< 19ω(G) for intersection graph G of a family of pseudo-discs, which improves a previous bound.  ...  Finally, note that one cannot hope for a similar bound, as in Corollary 2, on the number of edges in the intersection graph of a family of pseudo-circles in the plane.  ... 
arXiv:1312.1678v3 fatcat:riowdbdx6fephclx2hjtgb3kly

Coloring a set of touching strings

Louis Esperet, Daniel Gonçalves, Arnaud Labourel
2009 Electronic Notes in Discrete Mathematics  
We conjecture that also in this case, χ(F ) only depends on the maximum number of curves containing a given point of the plane.  ...  When F is a set of Jordan regions that may only intersect on their boundaries, and such that any point of the plane is contained in at most k regions, it can be proven that χ(F ) ≤ 3k/2 + o(k) since the  ...  This is not completely surprising: for instance, the best known upper bound on the chromatic number of K k -free intersection graphs of curves in the plane also depends on how many times two curves intersect  ... 
doi:10.1016/j.endm.2009.07.035 fatcat:6lrsctucozdtdm5zaqqm5gr46q

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, Bartosz Walczak
2013 Discrete & Computational Geometry  
However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane.  ...  Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number.  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s)  ... 
doi:10.1007/s00454-013-9534-9 fatcat:sccdix27orhd5p2ucvcdfchhvq

Disjointness graphs of short polygonal chains [article]

János Pach, Gábor Tardos, Géza Tóth
2021 arXiv   pre-print
It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G).  ...  In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G)≤(  ...  It was proved in [AG60] that the class of intersection graphs of axis-parallel rectangles in the plane is χ-bounded (see also [ChW20] ).  ... 
arXiv:2112.05991v1 fatcat:4rlll6bgqfbbjnhvhnluywryse

On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane

Seog-Jin Kim, Alexandr Kostochka, Kittikorn Nakprasit
2004 Electronic Journal of Combinatorics  
We show also that the chromatic number of every intersection graph $H$ of a family of homothetic copies of a fixed convex set in the plane with clique number $k$ is at most $6k-6$.  ...  Let $G$ be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane.  ...  In particular, the chromatic number and the list chromatic number of G do not exceed 3k − 2. The bound on degeneracy in Theorem 1 is sharp.  ... 
doi:10.37236/1805 fatcat:arkkmgvkr5aa7isirnggpbrgm4

Coloring Kk-free intersection graphs of geometric objects in the plane

Jacob Fox, János Pach
2012 European journal of combinatorics (Print)  
Erdős conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant.  ...  The intersection graph of a collection C of sets is the graph on the vertex set C, in which C 1 , C 2 ∈ C are joined by an edge if and only if C 1 ∩ C 2 ̸ = ∅.  ...  Acknowledgments The authors would like to thank Csaba Tóth for several helpful conversations concerning the content of this paper and for preparing the figure.  ... 
doi:10.1016/j.ejc.2011.09.021 fatcat:4ddm3ffsxvaileqak56flkjh5a

Coloring the complements of intersection graphs of geometric figures

Seog-Jin Kim, Kittikorn Nakprasit
2008 Discrete Mathematics  
We also study the chromatic number of the complement of the intersection graph of homothetic copies of a fixed convex body in R n .  ...  When n = 2, we show that (G) min{3 (G) − 2, 6 (G)}, where (G) is the size of the minimum dominating set of G. The bound on (G) 6 (G) is sharp.  ...  Acknowledgments The authors thank Alexandr V. Kostochka for his encouragement and helpful suggestions. The authors also thank Douglas B. West for his helpful comments on this paper.  ... 
doi:10.1016/j.disc.2007.08.072 fatcat:ru7osgm3a5frzayje5mb7d6tly

Outerstring graphs are χ-bounded [article]

Alexandre Rok, Bartosz Walczak
2018 arXiv   pre-print
An outerstring graph is an intersection graph of curves that lie in a common half-plane and have one endpoint on the boundary of that half-plane.  ...  We prove that the class of outerstring graphs is χ-bounded, which means that their chromatic number is bounded by a function of their clique number.  ...  Any bound on the chromatic number of intersection graphs of curves with clique number less than k implies a bound on the number of edges in k-quasi-planar graphs, as follows.  ... 
arXiv:1312.1559v4 fatcat:s2rkp67krrh55i4lwiejc4opje

Coloring kk-free intersection graphs of geometric objects in the plane

Jacob Fox, János Pach
2008 Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08  
Erdős conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant.  ...  More generally, we prove that for any t and k, the chromatic number of every K k -free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (c t  ...  Acknowledgement The authors would like to thank Csaba Tóth for several helpful conversations concerning the content of this paper and for preparing the figures.  ... 
doi:10.1145/1377676.1377735 dblp:conf/compgeom/FoxP08 fatcat:wztyvxxkdrgsbeox573bf6gure

Conflict-Free Coloring of Intersection Graphs [article]

Sándor P. Fekete, Phillip Keldenich
2017 arXiv   pre-print
We demonstrate that the intersection graph of n geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in Ω( n/ n) and in Ω(√( n)) for disks or squares  ...  Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs.  ...  The intersection graph of n convex objects in the plane may have conflict-free chromatic number in Ω(log n/ log log n).  ... 
arXiv:1709.03876v1 fatcat:ssd4liw3sncb5peerltvokuora

Geometric Achromatic and Pseudoachromatic Indices

O. Aichholzer, G. Araujo-Pardo, N. García-Colín, T. Hackl, D. Lara, C. Rubio-Montiel, J. Urrutia
2015 Graphs and Combinatorics  
A geometric graph is a graph drawn in the plane such that its vertices are points in general position, and its edges are straight-line segments.  ...  The pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex.  ...  Under this setting, the chromatic number of the graph shown in Figure 1 (a) is 4, while the chromatic number of the graph shown in Figure 1 (b) is 3.  ... 
doi:10.1007/s00373-015-1610-x fatcat:z6e76mpjtnat3goegj6x5duofy

Coloring triangle-free L-graphs with O(loglog n) colors [article]

Bartosz Walczak
2020 arXiv   pre-print
It is proved that triangle-free intersection graphs of n L-shapes in the plane have chromatic number O(loglog n).  ...  This improves the previous bound of O(log n) (McGuinness, 1996) and matches the known lower bound construction (Pawlik et al., 2013).  ...  By contrast, various classes of geometric intersection graphs are χ-bounded, which means that graphs of these classes have chromatic number bounded by some function of the clique number.  ... 
arXiv:2002.10755v1 fatcat:pb7mygmakvak3bu6veui76nwbq
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