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##
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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. ...

##
###
About equivalent interval colorings of weighted graphs

2009
*
Discrete Applied Mathematics
*

*For*

*a*fixed integer k,

*the*k-interval

*graph*

*coloring*problem (k-ICP) is to determine

*a*k-interval

*coloring*

*of*G with

*a*minimum

*number*

*of*conflicting edges. ... To this purpose, we define

*and*prove

*a*necessary

*and*sufficient condition

*for*

*the*

*equivalence*

*of*two k-interval

*colorings*. ...

*A*solution to

*the*classical k-

*coloring*problem is

*in*fact

*a*partition

*of*

*the*vertex set into k subsets called

*color*classes,

*and*

*the*total

*number*

*of*

*non*

*equivalent*k-

*colorings*is equal to

*the*

*number*

*of*...

##
###
Conditional Information Inequalities and Combinatorial Applications
[article]

2017
*
arXiv
*
pre-print

*The*second application is

*a*new method to prove

*lower*

*bounds*

*for*biclique coverings

*of*bipartite

*graphs*. ... y) there is at most one matching

*in*

*the*partition involving both x

*and*y; assume further that

*the*degree

*of*each left vertex is at least L

*and*

*the*degree

*of*each right vertex is at least R. ... ACKNOWLEDGMENT

*The*authors are grateful to Maxim Popov

*for*valuable comments on

*a*preliminary version

*of*this paper.

*The*authors thank anonymous referees

*of*IEEE Trans. Inform. ...

##
###
On the number of regular edge labelings

2014
*
Discrete Mathematics & Theoretical Computer Science
*

As an example

doi:10.46298/dmtcs.2085
fatcat:2dwaix5otbb3tlc36maswp3lfu
*of**the*wider applicability*of*this technique, we also improve*the*upper*bound*on*the**number**of*2-orientations*of**a*quadrangulation to O(1.87n). ... Our upper*bound*relies on*a*novel application*of*Shearer's entropy lemma. ... Acknowledgements We are grateful to*the*anonymous referees*for*their detailed comments. ...##
###
On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs

2018
*
Graphs and Combinatorics
*

We begin

doi:10.1007/s00373-018-1879-7
fatcat:zr4zggcjkbalvfroa77ylo2eaq
*a*general study*of**bounds*on M (k)*for*regular*graphs*, as well as*for**graphs*with varying list sizes. ...*For*k ≥ 3*property*M (k) has been studied only*for*multipartite*graphs*. Here we find*bounds*on M (k)*for**graphs*embedded on surfaces,*and*obtain new results on planar*graphs*. ... Then G has*property*M (k + l).*In**the*later sections, we obtain*a**lower**bound*on*the**average*degree*of*all UkLC*graphs*. ...##
###
On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs
[article]

2017
*
arXiv
*
pre-print

*For*k≥ 3

*property*M(k) has been studied only

*for*multipartite

*graphs*. Here we find

*bounds*on M(k)

*for*

*graphs*embedded on surfaces,

*and*obtain new results on planar

*graphs*. ... We begin

*a*general study

*of*

*bounds*on M(k)

*for*regular

*graphs*, as well as

*for*

*graphs*with varying list sizes. ... Then G has

*property*M (k + l).

*In*

*the*later sections, we obtain

*a*

*lower*

*bound*on

*the*

*average*degree

*of*all UkLC

*graphs*. ...

##
###
From χ- to χ_p-bounded classes
[article]

2021
*
arXiv
*
pre-print

On

arXiv:2009.02953v2
fatcat:woph6rwtvbdg7p3yjp5xxnsodq
*the*way, we generalize*a*result*of*Wood relating*the*chromatic*number**of**a**graph*to*the*star chromatic*number**of*its 1-subdivision. ... χ-*bounded*classes are studied here*in**the*context*of*star*colorings**and*more generally χ_p-*colorings*. ... Acknowledgements*The*authors would like to thank*the*anonymous referees*for*his corrections*and*suggestions. ...##
###
A Stress-Free Sum-of-Squares Lower Bound for Coloring
[article]

2021
*
arXiv
*
pre-print

To

arXiv:2105.07517v1
fatcat:eifhjitnuncmzf47bgzc7xpfsa
*the*best*of*our knowledge, this is*the*first*lower**bound**for**coloring*G(n,1/2)*for*even*a*single round strengthening*of**the*basic SDP*in*any SDP hierarchy. ... Our proof relies on*a*new variant*of*instance-preserving*non*-pointwise complete reduction within SoS from*coloring**a**graph*to finding large independent sets*in*it. ... Acknowledgements We thank Xinyu Wu*for*taking part*in*early stages*of*this research,*and**the*anonymous reviewers*for*providing valuable feedback. ...##
###
Oriented graph coloring

2001
*
Discrete Mathematics
*

An oriented k-

doi:10.1016/s0012-365x(00)00216-8
fatcat:qozv2mq72fgvlj2aur2lxib2aa
*coloring**of*an oriented*graph*G (that is*a*digraph with no cycle*of*length 2) is*a*partition*of*its vertex set into k subsets such that (i) no two adjacent vertices belong to*the*same subset ... We survey*the*main results that have been obtained on oriented*graph**colorings*. ...*The*gap between*the**lower**and**the*upper*bounds**for**the*oriented chromatic*number**of*planar*graphs*is thus still large*and*, despite many e orts, has not been reduced up to now. ...##
###
A Finite Graph Approach to the Probabilistic Hadwiger-Nelson Problem
[article]

2020
*
arXiv
*
pre-print

*The*contrapositive

*of*

*the*resulting

*bounds*lets us compute

*lower*

*bounds*on

*the*order

*of*

*non*k-

*colorable*unit-distance

*graphs*, improving

*bounds*produced by Pritikin

*and*

*the*Polymath16 project

*in*

*the*k = ... We define

*the*numerical badness

*of*

*a*given k-

*coloring*

*of*

*the*plane to be

*the*probability that

*a*randomly chosen unit-distance edge is monochromatic under

*the*

*coloring*,

*and*we provide

*lower*

*bounds*on

*the*... William Gasarch

*and*Dr. Clyde Kruskal

*of*

*the*University

*of*Maryland at College Park, as well as

*the*others involved

*in*coordinating

*the*REU program

*of*which this research was originally

*a*part. ...

##
###
Optimizing Regular Edge Labelings
[chapter]

2011
*
Lecture Notes in Computer Science
*

Furthermore, we give upper

doi:10.1007/978-3-642-18469-7_11
fatcat:vpg56oeryzfgvmvh6pxpjrs344
*and**lower**bounds*on*the**number**of*RELs. K. Buchin*and*B. Speckmann are supported by*the*Netherlands Organisation*for*Scientific Research (NWO) under project no. 639.022.707. ...*A*regular edge labeling (REL)*of*an irreducible triangulation G uniquely defines*a*rectangular dual*of*G. ...*For*this we need an upper*bound*δ on*the**number**of*neighbors*a*labeling can have: δ is*the**number**of*4-cycles*in**the*input*graph*. ...##
###
1-subdivisions, fractional chromatic number and Hall ratio
[article]

2020
*
arXiv
*
pre-print

*The*Hall ratio

*of*

*a*

*graph*G is

*the*maximum

*of*|V(H)|/alpha(H) over all subgraphs H

*of*G. Clearly,

*the*Hall ratio

*of*

*a*

*graph*is

*a*

*lower*

*bound*

*for*

*the*fractional chromatic

*number*. ... c > 0, every

*graph*

*of*sufficiently large

*average*degree contains as

*a*subgraph

*the*1-subdivision

*of*

*a*

*graph*

*of*fractional chromatic

*number*at least c. *

*For*every d > 0, there exists

*a*

*graph*G

*of*

*average*... Acknowledgments

*The*results on Hall ratio were inspired by discussions taking place during

*the*first Southwestern German Workshop on

*Graph*Theory

*in*Karlsruhe. ...

##
###
On Generalized Choice and Coloring Numbers

2019
*
Electronic Journal of Combinatorics
*

*A*well-known result

*of*Alon shows that

*the*

*coloring*

*number*

*of*

*a*

*graph*is

*bounded*by

*a*function

*of*its choosability. ... We explore this relationship

*in*

*a*more general setting with relaxed assumptions on

*color*classes, encoded by

*a*

*graph*parameter. ...

*The*

*bound*on

*the*

*coloring*

*number*(s ) from

*the*

*property*(CD) that we obtain

*in*

*the*proof

*of*Theorem 3

*of*course depends on both s

*and*p. ...

##
###
Upper bounds on the average number of colors in the non-equivalent colorings of a graph
[article]

2021
*
arXiv
*
pre-print

Two

arXiv:2105.01120v1
fatcat:5umi6mmjxnfunmxa67t3ejj4uu
*colorings*are*equivalent*if they induce*the*same partition*of**the*vertex set into*color*classes. Let*𝒜*(G) be*the**average**number**of**colors**in**the**non*-*equivalent**colorings**of**a**graph*G. ... We give*a*general upper*bound*on*𝒜*(G) that is valid*for*all*graphs*G*and**a*more precise one*for**graphs*G*of*order n*and*maximum degree Δ(G)∈{1,2,n-2}. ... Acknowledgments*The*authors thank Julien Poulain*for*his precious help*in*optimizing our programs allowing us to check our conjectures on*a*large*number**of**graphs*. ...##
###
An optimal lower bound on the number of variables for graph identification

1989
*
30th Annual Symposium on Foundations of Computer Science
*

We thus settle

doi:10.1109/sfcs.1989.63543
dblp:conf/focs/CaiFI89
fatcat:cqgtmai7ibdppprm3pd7umjh4y
*a*long-standing open problem. Previously it was an open question whether or not 4 variables su ce. Our*lower**bound*remains true over*a*set*of**graphs**of**color*class size 4. ...*In*this paper we show that (n) variables are needed*for*rst-order logic with counting to identify*graphs*on n vertices.*The*k-variable language with counting is*equivalent*to*the*(k ? ... Thanks to Laci Babai*for*informing us about*the*status*of**the*research on*the*W-L method*in**the*Soviet Union. ...
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