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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

About equivalent interval colorings of weighted graphs

Mathieu Bouchard, Mirjana Čangalović, Alain Hertz
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  ... 
doi:10.1016/j.dam.2009.04.015 fatcat:7pa2infuv5b45kdoxeuhxztllq

Conditional Information Inequalities and Combinatorial Applications [article]

Tarik Kaced, Andrei Romashchenko, Nikolay Vereshchagin
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.  ... 
arXiv:1501.04867v4 fatcat:5strcot2fzbzzbondsv4xopkdy

On the number of regular edge labelings

Kevin Buchin, Bettina Speckmann, Sander Verdonschot
2014 Discrete Mathematics & Theoretical Computer Science  
As an example 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.  ... 
doi:10.46298/dmtcs.2085 fatcat:2dwaix5otbb3tlc36maswp3lfu

On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs

M. Abdolmaleki, J. P. Hutchinson, S. Gh. Ilchi, E. S. Mahmoodian, N. Matsumoto, M. A. Shabani
2018 Graphs and Combinatorics  
We begin 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.  ... 
doi:10.1007/s00373-018-1879-7 fatcat:zr4zggcjkbalvfroa77ylo2eaq

On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs [article]

M. Abdolmaleki, J. P. Hutchinson, S. Gh. Ilchi, E. S. Mahmoodian, M. A. Shabani
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.  ... 
arXiv:1705.07434v1 fatcat:kb7pzk5o3nd3nf4eedxikfcuei

From χ- to χ_p-bounded classes [article]

Y. Jiang, J. Nesetril, P. Ossona de Mendez
2021 arXiv   pre-print
On 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.  ... 
arXiv:2009.02953v2 fatcat:woph6rwtvbdg7p3yjp5xxnsodq

A Stress-Free Sum-of-Squares Lower Bound for Coloring [article]

Pravesh K. Kothari, Peter Manohar
2021 arXiv   pre-print
To 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.  ... 
arXiv:2105.07517v1 fatcat:eifhjitnuncmzf47bgzc7xpfsa

Oriented graph coloring

Eric Sopena
2001 Discrete Mathematics  
An oriented k-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.  ... 
doi:10.1016/s0012-365x(00)00216-8 fatcat:qozv2mq72fgvlj2aur2lxib2aa

A Finite Graph Approach to the Probabilistic Hadwiger-Nelson Problem [article]

Haydn Gwyn, Jacob Stavrianos
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.  ... 
arXiv:2008.07987v1 fatcat:5mm7nqdtsnf7xlyyoqseetppla

Optimizing Regular Edge Labelings [chapter]

Kevin Buchin, Bettina Speckmann, Sander Verdonschot
2011 Lecture Notes in Computer Science  
Furthermore, we give upper 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.  ... 
doi:10.1007/978-3-642-18469-7_11 fatcat:vpg56oeryzfgvmvh6pxpjrs344

1-subdivisions, fractional chromatic number and Hall ratio [article]

Zdeněk Dvořák and Patrice Ossona de Mendez and Hehui Wu
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.  ... 
arXiv:1812.07327v2 fatcat:vqgotw2mvff4hn2scvsbna4pka

On Generalized Choice and Coloring Numbers

Zdenēk Dvořák, Jakub Pekárek, Jean-Sébastien Sereni
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.  ... 
doi:10.37236/7693 fatcat:ta76rmkt3rdm7ly5zwzatu3gdu

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

Alain Hertz, Hadrien Mélot, Sébastien Bonte, Gauvain Devillez, Pierre Hauweele
2021 arXiv   pre-print
Two 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.  ... 
arXiv:2105.01120v1 fatcat:5umi6mmjxnfunmxa67t3ejj4uu

An optimal lower bound on the number of variables for graph identification

J.-Y. Cai, M. Furer, N. Immerman
1989 30th Annual Symposium on Foundations of Computer Science  
We thus settle 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.  ... 
doi:10.1109/sfcs.1989.63543 dblp:conf/focs/CaiFI89 fatcat:cqgtmai7ibdppprm3pd7umjh4y
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