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### Exact Algorithms for Coloring Graphs While Avoiding Monochromatic Cycles [chapter]

Fabrice Talla Nobibon, Cor Hurkens, Roel Leus, Frits C. R. Spieksma
<span title="">2010</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;
of vertices that can be colored using two colors while avoiding monochromatic cycles.  ...  We present three exact algorithms, namely an integer-programming algorithm based on cycle identification, a backtracking algorithm, and a branch-and-check algorithm.  ...  number of vertices that can be colored using two colors while avoiding monochromatic cycles.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-14355-7_24">doi:10.1007/978-3-642-14355-7_24</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r6gqayhmj5dtzcum37ra4jtkbq">fatcat:r6gqayhmj5dtzcum37ra4jtkbq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170816064647/http://orbi.ulg.ac.be/bitstream/2268/97782/1/coloringgraph_report.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/45/ba/45ba660f56215a8d0d5b965aa0d111e2fb569392.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-14355-7_24"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### Coloring Graphs Using Two Colors While Avoiding Monochromatic Cycles

Fabrice Talla Nobibon, Cor A. J. Hurkens, Roel Leus, Frits C. R. Spieksma
<span title="">2012</span> <i title="Institute for Operations Research and the Management Sciences (INFORMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ko3yoz32wja7tmqwbgqqjmfyxa" style="color: black;">INFORMS journal on computing</a> </i> &nbsp;
of vertices that can be colored using two colors while avoiding monochromatic cycles.  ...  We present three exact algorithms-namely, an integer-programming algorithm based on cycle identification, a backtracking algorithm, and a branch-and-check algorithm.  ...  Acknowledgments The authors thank the referees and the associate editor for their constructive comments, and in particular for suggesting the alternative branching strategy.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1287/ijoc.1110.0466">doi:10.1287/ijoc.1110.0466</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/k72723y4hfdl7dfofwfcgvevpu">fatcat:k72723y4hfdl7dfofwfcgvevpu</a> </span>
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### Exact defective colorings of graphs [article]

James Cumberbatch, Juho Lauri, Christodoulos Mitillos
<span title="2021-09-11">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also give polynomial-time algorithms for finding certain types of exact (k,d)-colorings in cactus graphs and block graphs.  ...  We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs.  ...  Algorithms for exact d-defective coloring structured graphs In this section, we give positive algorithmic results for finding exact (k, d)-colorings of certain structured graphs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2109.05255v1">arXiv:2109.05255v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bzge2rr4srcdjhbp3lq2sz4jma">fatcat:bzge2rr4srcdjhbp3lq2sz4jma</a> </span>
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### On the complexity of edge-colored subgraph partitioning problems in network optimization

Xiaoyan Zhang, Zan-Bo Zhang, Hajo Broersma, Xuelian Wen
<span title="2016-07-01">2016</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;
Moreover, for the weighted version of the minimum monochromatic clique partition problem on monochromatic-diamond-free graphs, we derive an approximation algorithm with (tight) approximation guarantee  ...  Given a (not necessarily properly) edge-colored graph \$G=(V,E)\$, a subgraph \$H\$ is said to be <i>monochromatic</i> if all its edges have the same color, and called <i>multicolored</i> if all its edges  ...  Acknowledgements The authors would like to thank the anonymous referees for their valuable suggestions and comments that have helped a lot to improve the quality and presentation of the paper.  ...
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### New directions in Ramsey theory

<span title="2021-03-11">2021</span> <i title="Shahin Digital Publisher"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bdwdrpk3ire47o4q6hmknhdqya" style="color: black;">Discrete Mathematics Letters</a> </i> &nbsp;
To see that R(3, 3) = 6 is quite easy, for in any red-blue coloring of the complete graph K 6 , every vertex is incident with at least three edges of the same color, say vv 1 , vv 2 , and vv 3 are three  ...  (Ramsey's Theorem) For every two positive integers s and t, there exists a positive integer N such that for every red-blue coloring of K N , there is a complete subgraph K s all of whose edges are colored  ...  From this, it follows that every red-blue coloring of K 5,5 results in a monochromatic C 4 , while there exists a red-blue coloring of K 4,5 that avoids a monochromatic C 4 .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.47443/dml.2021.s110">doi:10.47443/dml.2021.s110</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/b2c65o5bg5cdvhbxaefcsyssme">fatcat:b2c65o5bg5cdvhbxaefcsyssme</a> </span>
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### On Some Three-Color Ramsey Numbers for Paths [article]

Janusz Dybizbański, Tomasz Dzido, Stanisław Radziszowski
<span title="2015-07-05">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
a monochromatic copy of G_i in color i, for some 1 ≤ i ≤ 3.  ...  For graphs G_1, G_2, G_3, the three-color Ramsey number R(G_1, G_2, G_3) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then it contains  ...  graph of order n, K n , then it contains a monochromatic copy of G i in color i, for some 1 ≤ i ≤ k.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1207.5851v3">arXiv:1207.5851v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v2ksvzoms5esvf2zpiqikq5yom">fatcat:v2ksvzoms5esvf2zpiqikq5yom</a> </span>
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### On some three-color Ramsey numbers for paths

Janusz Dybizbański, Tomasz Dzido, Stanisław Radziszowski
<span title="">2016</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;
it contains a monochromatic copy of G i in color i, for some 1 ≤ i ≤ 3.  ...  For graphs G 1 , G 2 , G 3 , the three-color Ramsey number R(G 1 , G 2 , G 3 ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then  ...  graph of order n, K n , then it contains a monochromatic copy of G i in color i, for some 1 ≤ i ≤ k.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2015.11.002">doi:10.1016/j.dam.2015.11.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d64nqkbq7zag3ikhk2ripmvq4y">fatcat:d64nqkbq7zag3ikhk2ripmvq4y</a> </span>
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### On-line Ramsey numbers for paths and stars

J. A. Grytczuk, H. A. Kierstead, P. Prałat
<span title="2008-01-01">2008</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;
We determine exact values of r(H) for a few short paths and obtain a general upper bound r(Pn) ≤ 4n −7.  ...  The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible.  ...  For given graphs G and H we write G → H if there is a monochromatic copy of H in any red-blue coloring of the edges of G.  ...
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### Anti-coordination Games and Stable Graph Colorings [chapter]

Jeremy Kun, Brian Powers, Lev Reyzin
<span title="">2013</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 prove the decision problem for non-strict equilibria in directed graphs is NP-hard.  ...  Motivated by understanding non-strict and strict pure strategy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs.  ...  We thank György Turán for helpful discussions. References  ...
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### Anti-Coordination Games and Stable Graph Colorings [article]

Jeremy Kun, Brian Powers, Lev Reyzin
<span title="2013-08-14">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove the decision problem for non-strict equilibria in directed graphs is NP-hard.  ...  Motivated by understanding non-strict and strict pure strategy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs.  ...  Acknowledgements We thank György Turán for helpful discussions. References  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1308.3258v1">arXiv:1308.3258v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tmtr2bfrw5ef5jpci5kjb7j2ci">fatcat:tmtr2bfrw5ef5jpci5kjb7j2ci</a> </span>
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### Turán and Ramsey-type results for unavoidable subgraphs [article]

Alp Müyesser, Michael Tait
<span title="2020-04-19">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of edges in an n-vertex ε-balanced graph which avoids all of F.  ...  In this paper, we consider this Turán function for several classes of edge-colored graphs, we show that the Ramsey function is linear for bounded degree graphs, and we prove a theorem that gives a relationship  ...  of graphs avoiding nonmonochromatic K 4 .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2004.07147v2">arXiv:2004.07147v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xaj5y5jpifeyrfqkskvj3owoea">fatcat:xaj5y5jpifeyrfqkskvj3owoea</a> </span>
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### Acyclic edge colorings of graphs

Noga Alon, Benny Sudakov, Ayal Zaks
<span title="">2001</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G.  ...  This can be achieved by recoloring one edge from each bichromatic cycle using one additional color, while avoiding recoloring any pair of edges which are incident or at distance 1 from each other (similar  ...  We have found another proof for this case, which also yields a polynomial algorithm for acyclically coloring the edges of a graph of maximum degree 3 using ®ve colors.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.1010">doi:10.1002/jgt.1010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4wd54gnawrbopebcl2quvqzdz4">fatcat:4wd54gnawrbopebcl2quvqzdz4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20060901114811/http://www.math.princeton.edu/~bsudakov/acyclic.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/ab/40/ab4061a49116b93ae36b2506badb66d43cc5be62.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.1010"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

### The Parameterized Complexity of Some Minimum Label Problems [chapter]

Michael R. Fellows, Jiong Guo, Iyad A. Kanj
<span title="">2010</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;
For the background and terminologies on graphs, we refer the reader to West [16], and for that on parameterized complexity, we refer the reader to Downey and Fellows' book [7] .  ...  The problems under consideration are: Minimum Label Spanning Tree (MLST), Minimum Label Hamiltonian Cycle (MLHC) (where Π is the property of being a Hamiltonian cycle), Minimum Label Cut (MLC), Minimum  ...  Some of the previous work, however, dealt with developing exact algorithms for these problems. For example, Broersma et al.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-11409-0_8">doi:10.1007/978-3-642-11409-0_8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oq56o3we5jckpjnlmb7wfapt34">fatcat:oq56o3we5jckpjnlmb7wfapt34</a> </span>
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### Restricted size Ramsey number for P_3 versus cycles [article]

Joanna Cyman, Tomasz Dzido
<span title="2018-11-20">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let F, G and H be simple graphs. We say F → (G, H) if for every 2-coloring of the edges of F there exists a monochromatic G or H in F.  ...  In this paper we determine previously unknown restricted size Ramsey numbers r^*(P_3, C_n) for 7 ≤ n ≤ 12. We also give new upper bound r^*(P_3, C_n) ≤ 2n-2 for even n ≥ 8.  ...  We would like to thank the student of the University of Gdańsk Maciej Godek for the independent performance of some computer experiments that confirmed the correctness of the results contained in the article  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1706.08134v2">arXiv:1706.08134v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pquwjhxnxvacbkerhuztmtuhbq">fatcat:pquwjhxnxvacbkerhuztmtuhbq</a> </span>
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### The parameterized complexity of some minimum label problems

Michael R. Fellows, Jiong Guo, Iyad Kanj
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p6ovb2qpkfenhmb7mcksobrcxq" style="color: black;">Journal of computer and system sciences (Print)</a> </i> &nbsp;
Proving that some of these problems are FPT requires interesting algorithmic methods that we develop in this paper.  ...  We study the parameterized complexity of several minimum label graph problems, in which we are given an undirected graph whose edges are labeled, and a property Π , and we are asked to find a subset of  ...  Acknowledgment We would like to thank the anonymous referee for the valuable comments and suggestions that helped improve the quality and the presentation of the paper.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcss.2010.02.012">doi:10.1016/j.jcss.2010.02.012</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/umluypvknfbexcunaghu3pkdce">fatcat:umluypvknfbexcunaghu3pkdce</a> </span>
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