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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
<span title="2021-05-03">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2105.01120v1">arXiv:2105.01120v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5umi6mmjxnfunmxa67t3ejj4uu">fatcat:5umi6mmjxnfunmxa67t3ejj4uu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210506025312/https://arxiv.org/pdf/2105.01120v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d3/84/d384e897a6452531fe925bd3ad609d37e5fdc8df.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2105.01120v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

On the number of regular edge labelings

Kevin Buchin, Bettina Speckmann, Sander Verdonschot
<span title="2014-10-20">2014</span> <i title="Centre pour la Communication Scientifique Directe (CCSD)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aagtqr2vajamvduhte7kigeygi" style="color: black;">Discrete Mathematics &amp; Theoretical Computer Science</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.46298/dmtcs.2085">doi:10.46298/dmtcs.2085</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2dwaix5otbb3tlc36maswp3lfu">fatcat:2dwaix5otbb3tlc36maswp3lfu</a> </span>
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About equivalent interval colorings of weighted graphs

Mathieu Bouchard, Mirjana Čangalović, Alain Hertz
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
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.  ...  Given a graph G = (V , E) with strictly positive integer weights ω i on the vertices i ∈ V , a k-interval coloring of G is a function I that assigns an interval I(i) ⊆ {1, . . . , k} of ω i consecutive  ...  Upper bounds on the interval chromatic number χ int (G) are studied in [5] , while general upper and lower bounds on χ int (G) are given in [15] when vertices possibly have forbidden colors.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2009.04.015">doi:10.1016/j.dam.2009.04.015</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7pa2infuv5b45kdoxeuhxztllq">fatcat:7pa2infuv5b45kdoxeuhxztllq</a> </span>
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An algorithm for finding a maximum clique in a graph

David R Wood
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z7xgofryqjgkxjlwzlky45v4li" style="color: black;">Operations Research Letters</a> </i> &nbsp;
This paper introduces a branch-and-bound algorithm for the maximum clique problem which applies existing clique ÿnding and vertex coloring heuristics to determine lower and upper bounds for the size of  ...  Computational results on a variety of graphs indicate the proposed procedure in most instances outperforms leading algorithms. c 1997 Elsevier Science B.V. All rights reserved.  ...  Thanks also to Christopher Pickett for the use of his computer.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0167-6377(97)00054-0">doi:10.1016/s0167-6377(97)00054-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rl7geoxopfenpaaz55ioqweimy">fatcat:rl7geoxopfenpaaz55ioqweimy</a> </span>
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The clique-partitioning problem

J. Bhasker, Tariq Samad
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nkrwe4pmozafvnd72yxufztpku" style="color: black;">Computers and Mathematics with Applications</a> </i> &nbsp;
We study the problem of clique-partitioning a graph. We prove a new generai upper bound resuit on the number of clique-partitions.  ...  Since the clique-partitioning of a graph is equivalent to the coioring of the complement of the graph, any coloring algorithm can aiso be used to ciiqu~partition a graph.  ...  The upper bound on 0 is symbolized as eupper. n represents the number of vertices in the graph and e the number of edges.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0898-1221(91)90001-k">doi:10.1016/0898-1221(91)90001-k</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z2wd5dyupraexo5x2uxwmyyw5q">fatcat:z2wd5dyupraexo5x2uxwmyyw5q</a> </span>
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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
<span title="">2009</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
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  ...  Graph coloring produces color classes with each class representing an independent set of vertices.  ...  Saeed Abu-Nimeh for several valuable discussions and Ilteris Murat Derici for his initial work on the SL algorithm and RGG.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-03417-6_4">doi:10.1007/978-3-642-03417-6_4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hbvpocsncbcwhmic75suu76asq">fatcat:hbvpocsncbcwhmic75suu76asq</a> </span>
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Upper Bounds on Sets of Orthogonal Colorings of Graphs [article]

Serge C. Ballif
<span title="2012-11-29">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Two n-colorings of a graph are said to be orthogonal if whenever two vertices share a color in one coloring they have distinct colors in the other coloring.  ...  latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.  ...  Acknowledgements 465 The author is indebted to Yu Caro for pointing out several relevant references.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1110.2237v3">arXiv:1110.2237v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wz5a3m2tgrhxnjh7imofwe2lsy">fatcat:wz5a3m2tgrhxnjh7imofwe2lsy</a> </span>
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Another approach to non-repetitive colorings of graphs of bounded degree [article]

Matthieu Rosenfeld
<span title="2020-06-23">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The application we provide in this paper are upper bounds for graphs of maximal degree at most Δ: a minor improvement on the upper-bound of the non-repetitive number, a 4.25Δ +o(Δ) upper-bound on the weak  ...  We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive numbers to the maximal degree of a graph.  ...  Acknowledgement I wish to thank Gwenaël Joret and William Lochet for comments on earlier drafts.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.09094v2">arXiv:2006.09094v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5auztmurafgepdsniotsaolrci">fatcat:5auztmurafgepdsniotsaolrci</a> </span>
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Another Approach to Non-Repetitive Colorings of Graphs of Bounded Degree

Matthieu Rosenfeld
<span title="2020-09-04">2020</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive chromatic numbers to the maximal degree of a graph.  ...  The applications we provide in this paper are upper bounds for graphs of maximal degree at most $\Delta$: a minor improvement on the upper-bound of the non-repetitive chromatic number, a $4.25\Delta +o  ...  Acknowledgment I wish to thank Gwenaël Joret and William Lochet for comments on earlier drafts.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/9667">doi:10.37236/9667</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7xfj7avwzzh35luiqxpa6nromi">fatcat:7xfj7avwzzh35luiqxpa6nromi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200916135309/https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p43/8161" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/68/77/687771726f37c527c530e70a1a594601e793cbc5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/9667"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Two Lagrangian Relaxation Based Heuristics for Vertex Coloring Problem

Ali İrfan MAHMUTOĞULLARI
<span title="2020-06-25">2020</span> <i title="Cumhuriyet University Sciences Journal"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/thxymt26tjhcnnqnqwc3esnvua" style="color: black;">Cumhuriyet Science Journal</a> </i> &nbsp;
Vertex coloring problem is a well-known NP-Hard problem where the objective is to minimize the number of colors used to color vertices of a graph ensuring that adjacent vertices cannot have same color.  ...  In this paper, we first discuss existing mathematical formulations of the problem and then consider two different heuristics, namely HEUR-RA and HEUR-RC, based on Lagrangian relaxation of adjacency and  ...  The vertex coloring problem (VCP) is to determine chromatic number, denoted as ( ), of a given graph where chromatic number of a graph is the smallest number of colors to color it.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.17776/csj.628518">doi:10.17776/csj.628518</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/keigx6acqfhyppyklqt2km2xhu">fatcat:keigx6acqfhyppyklqt2km2xhu</a> </span>
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Oriented graph coloring

Eric Sopena
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We survey the main results that have been obtained on oriented graph colorings.  ...  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  ...  Such a k-coloring can be equivalently regarded as a homomorphism of H to the complete graph K k on k-vertices.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(00)00216-8">doi:10.1016/s0012-365x(00)00216-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qozv2mq72fgvlj2aur2lxib2aa">fatcat:qozv2mq72fgvlj2aur2lxib2aa</a> </span>
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Optimizing Regular Edge Labelings [chapter]

Kevin Buchin, Bettina Speckmann, Sander Verdonschot
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-18469-7_11">doi:10.1007/978-3-642-18469-7_11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vpg56oeryzfgvmvh6pxpjrs344">fatcat:vpg56oeryzfgvmvh6pxpjrs344</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190220195434/http://pdfs.semanticscholar.org/28c3/44c50b2bd5df7a3c5aa149e357c6e1171f9a.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/28/c3/28c344c50b2bd5df7a3c5aa149e357c6e1171f9a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-18469-7_11"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Clause Learning and New Bounds for Graph Coloring

Emmanuel Hebrard, George Katsirelos
<span title="">2019</span> <i title="International Joint Conferences on Artificial Intelligence Organization"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vfwwmrihanevtjbbkti2kc3nke" style="color: black;">Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence</a> </i> &nbsp;
The combination of these techniques in a branch- and-bound search outperforms Dsatur and other SAT-based approaches on standard benchmarks both for finding upper bounds and for proving lower bounds.  ...  Branching on whether two neighbors get the same color yields a symmetry-free tree with complete graphs as leaves, which correspond to colorings of the original graph.  ...  We show the ratio of instances for which a proof of optimality was found ('opt'), as well as the average upper bound ('ub') and lower bound ('lb'), for every method.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.24963/ijcai.2019/856">doi:10.24963/ijcai.2019/856</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/ijcai/HebrardK19.html">dblp:conf/ijcai/HebrardK19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ypjb6xlx2racxapzds63xgonla">fatcat:ypjb6xlx2racxapzds63xgonla</a> </span>
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The maximum k-colorable subgraph problem and related problems [article]

Renata Sotirov, Olga Kuryatnikova, Juan Vera
<span title="2021-02-11">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also show how to exploit invariance under permutations of the subsets for other partition problems and how to use the MkCS problem to derive bounds on the chromatic number of a graph.  ...  The maximum k-colorable subgraph (MkCS) problem is to find an induced k-colorable subgraph with maximum cardinality in a given graph.  ...  Our lower bounds on the chromatic number of a graph are competitive with bounds from the literature. Acknowledgment.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.09644v3">arXiv:2001.09644v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2sdebbgc6zg5pdaiba6me3aoqm">fatcat:2sdebbgc6zg5pdaiba6me3aoqm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210213153907/https://arxiv.org/pdf/2001.09644v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/69/63/696300dae2eb8904fd2998e15855798ddebdbebf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.09644v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

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
<span title="2018-03-06">2018</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yxooi3wjmbgqtbp7l4evjeuj4i" style="color: black;">Graphs and Combinatorics</a> </i> &nbsp;
We begin a general study of bounds on M (k) for regular graphs, as well as for graphs with varying list sizes.  ...  A graph G is called uniquely k-list colorable (UkLC) if there exists a list of colors on its vertices, say L = {S v | v ∈ V (G)}, each of size k, such that there is a unique proper list coloring of G from  ...  In the later sections, we obtain a lower bound on the average degree of all UkLC graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00373-018-1879-7">doi:10.1007/s00373-018-1879-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zr4zggcjkbalvfroa77ylo2eaq">fatcat:zr4zggcjkbalvfroa77ylo2eaq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190307195814/http://pdfs.semanticscholar.org/e63e/4583f634f0b38a821f61fd1800b3fa31f696.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e6/3e/e63e4583f634f0b38a821f61fd1800b3fa31f696.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00373-018-1879-7"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>
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