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

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] . ...

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

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. ...

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

2021
*
arXiv
*
pre-print

We prove that there are

arXiv:2011.14174v2
fatcat:hnu3542pu5a5lkefd7zyqzljpm
*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. ...##
###
Note on the number of edges in families with linear union-complexity
[article]

2014
*
arXiv
*
pre-print

We give a simple argument showing that

arXiv:1312.1678v3
fatcat:riowdbdx6fephclx2hjtgb3kly
*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*. ...##
###
Coloring a set of touching strings

2009
*
Electronic Notes in Discrete Mathematics
*

We conjecture that also

doi:10.1016/j.endm.2009.07.035
fatcat:6lrsctucozdtdm5zaqqm5gr46q
*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*...##
###
Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

2013
*
Discrete & Computational Geometry
*

However, until very recently, no such construction was known for

doi:10.1007/s00454-013-9534-9
fatcat:sccdix27orhd5p2ucvcdfchhvq
*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) ...##
###
Disjointness graphs of short polygonal chains
[article]

2021
*
arXiv
*
pre-print

It is known that

arXiv:2112.05991v1
fatcat:4rlll6bgqfbbjnhvhnluywryse
*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] ). ...##
###
On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane

2004
*
Electronic Journal of Combinatorics
*

We show also that

doi:10.37236/1805
fatcat:arkkmgvkr5aa7isirnggpbrgm4
*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. ...##
###
Coloring Kk-free intersection graphs of geometric objects in the plane

2012
*
European journal of combinatorics (Print)
*

Erdős conjectured that

doi:10.1016/j.ejc.2011.09.021
fatcat:4ddm3ffsxvaileqak56flkjh5a
*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. ...##
###
Coloring the complements of intersection graphs of geometric figures

2008
*
Discrete Mathematics
*

We also study

doi:10.1016/j.disc.2007.08.072
fatcat:ru7osgm3a5frzayje5mb7d6tly
*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. ...##
###
Outerstring graphs are χ-bounded
[article]

2018
*
arXiv
*
pre-print

An outerstring

arXiv:1312.1559v4
fatcat:s2rkp67krrh55i4lwiejc4opje
*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. ...##
###
Coloring kk-free intersection graphs of geometric objects in the plane

2008
*
Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08
*

Erdős conjectured that

doi:10.1145/1377676.1377735
dblp:conf/compgeom/FoxP08
fatcat:wztyvxxkdrgsbeox573bf6gure
*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. ...##
###
Conflict-Free Coloring of Intersection Graphs
[article]

2017
*
arXiv
*
pre-print

We demonstrate that

arXiv:1709.03876v1
fatcat:ssd4liw3sncb5peerltvokuora
*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). ...##
###
Geometric Achromatic and Pseudoachromatic Indices

2015
*
Graphs and Combinatorics
*

A geometric

doi:10.1007/s00373-015-1610-x
fatcat:z6e76mpjtnat3goegj6x5duofy
*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. ...##
###
Coloring triangle-free L-graphs with O(loglog n) colors
[article]

2020
*
arXiv
*
pre-print

It is proved that triangle-free

arXiv:2002.10755v1
fatcat:pb7mygmakvak3bu6veui76nwbq
*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*. ...
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