Filters








3,085 Hits in 10.1 sec

Bipartite induced density in triangle-free graphs [article]

Wouter Cames van Batenburg, Rémi de Joannis de Verclos, Ross J. Kang, François Pirot
<span title="2020-04-12">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that any triangle-free graph on n vertices with minimum degree at least d contains a bipartite induced subgraph of minimum degree at least d^2/(2n).  ...  Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of n/d and (2+o(1))√(n/log n) as n→∞. This is sharp up to constant factors.  ...  We thank Matthew Kwan, Benny Sudakov and Tuan Tran for informing us of their independent, concurrent work and of the paper of Poljak and Tuza.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1808.02512v3">arXiv:1808.02512v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lvzw7566tbgjnhmdt26lmubgty">fatcat:lvzw7566tbgjnhmdt26lmubgty</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200415113551/https://arxiv.org/pdf/1808.02512v3.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] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1808.02512v3" 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>

Coloring dense graphs via VC-dimension [article]

Tomasz Łuczak, Stéphan Thomassé
<span title="2010-07-09">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show how the usual VC-dimension gives a short proof of the fact that triangle-free graphs with minimum degree at least n/3 have bounded chromatic number, where n is the number of vertices.  ...  In other words, one can find H-free graphs with unbounded chromatic number and minimum degree arbitrarily close to n/3. These H-free graphs are derived from a construction of Hajnal.  ...  This work was done while the authors were visiting IPAM.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1007.1670v1">arXiv:1007.1670v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q4m3klqqivcgfogfwnmeb35qvu">fatcat:q4m3klqqivcgfogfwnmeb35qvu</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1007.1670/1007.1670.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> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a3/97/a3971b0e4154850156cda91d72621a309b214213.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1007.1670v1" 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>

Bipartite Induced Density in Triangle-Free Graphs

Wouter Cames van Batenburg, Rémi De Joannis de Verclos, Ross J. Kang, François Pirot
<span title="2020-05-29">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;
Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$.  ...  We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$.  ...  We thank Matthew Kwan, Benny Sudakov and Tuan Tran for informing us of their independent, concurrent work and of the paper of Poljak and Tuza.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/8650">doi:10.37236/8650</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gkhy6dapx5c7hh4r7ql5hqekum">fatcat:gkhy6dapx5c7hh4r7ql5hqekum</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200603222336/https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p34/8090" 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/9f/2b/9f2bed15699ecc8bb329c5e48ab1949a8cbe0ca9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/8650"> <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>

Graphs with Large Girth Not Embeddable in the Sphere

PIERRE CHARBIT, STÉPHAN THOMASSÉ
<span title="2007-07-20">2007</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/domxx2wewzae3e2m2ziqaqbyhm" style="color: black;">Combinatorics, probability &amp; computing</a> </i> &nbsp;
We discuss then the case of triangle-free graphs with linear minimum degree.  ...  In 1978, Larman [4] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed p 8/3.  ...  So the last question is to find what could be the largest chromatic number of a triangle-free graph with minimum degree n/3. The answer can be anything between four and infinity.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548307008528">doi:10.1017/s0963548307008528</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mugadslepzdorm2fgorvwskhfu">fatcat:mugadslepzdorm2fgorvwskhfu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190314061911/https://core.ac.uk/download/pdf/47117641.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/52/a1/52a15d3672c7d619cface4f2d7e1023498c9b97d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548307008528"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> cambridge.org </button> </a>

Bounds for the smallest k-chromatic graphs of given girth [article]

Geoffrey Exoo, Jan Goedgebeur
<span title="2019-03-05">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let n_g(k) denote the smallest order of a k-chromatic graph of girth at least g. We consider the problem of determining n_g(k) for small values of k and g.  ...  After giving an overview of what is known about n_g(k), we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction  ...  Most of the computations were carried out using the Stevin Supercomputer Infrastructure at Ghent University and the computers at the Computer Science labs at Indiana State University.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.06713v4">arXiv:1805.06713v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/l6jkopozsrbnhmzfjyct4256ge">fatcat:l6jkopozsrbnhmzfjyct4256ge</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200828045550/https://arxiv.org/pdf/1805.06713v4.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/b3/53/b3539db445f8bda398aca0adb14e6e1fc1e45e06.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.06713v4" 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>

Bounds for the smallest $k$-chromatic graphs of given girth

Geoffrey Exoo, Jan Goedgebeur
<span title="">2019</span> <i title="Episciences.org"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aagtqr2vajamvduhte7kigeygi" style="color: black;">Discrete Mathematics &amp; Theoretical Computer Science</a> </i> &nbsp;
Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$.  ...  After giving an overview of what is known about $n_g(k)$, we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction  ...  Most of the computations were carried out using the Stevin Supercomputer Infrastructure at Ghent University and the computers at the Computer Science labs at Indiana State University.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.23638/dmtcs-21-3-9">doi:10.23638/dmtcs-21-3-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qe5pqaijwren3jmuebtcahaypu">fatcat:qe5pqaijwren3jmuebtcahaypu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200218033217/https://dmtcs.episciences.org/5259/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/64/70/64703c754260f16e07fe0e1c20302e57f330b807.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.23638/dmtcs-21-3-9"> <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>

Dense graphs with small clique number

Wayne Goddard, Jeremy Lyle
<span title="2010-12-16">2010</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;
We consider the structure of K r -free graphs with large minimum degree, and show that such graphs with minimum degree δ > (2r − 5)n/(2r − 3) are homomorphic to the join K r−3 ∨ H where H is a triangle-free  ...  In particular this allows us to generalize results from triangle-free graphs and show that K r -free graphs with such minimum degree have chromatic number at most r + 1.  ...  Acknowledgments We would like to thank the referees for several helpful comments, and for pointing us to [4] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.20505">doi:10.1002/jgt.20505</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/66c3rveikza4noctoymttzav4e">fatcat:66c3rveikza4noctoymttzav4e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809033649/https://people.cs.clemson.edu/~goddard/papers/denseNoClique.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/22/9e/229ecb92b168700513ecadb34731ed27a34611d0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.20505"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

Structure and colour in triangle-free graphs [article]

N. R. Aravind, Stijn Cambie, Wouter Cames van Batenburg, Rémi de Joannis de Verclos, Ross J. Kang, Viresh Patel
<span title="2020-03-15">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number χ contains a rainbow independent set of size 1/2χ.  ...  and induced H-free graph has chromatic number at most c D/log D.  ...  We are also grateful to Gwenaël Joret for his help in identifying a subtlety in an earlier version of this work.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1912.13328v2">arXiv:1912.13328v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/32k2aqze7jbizg3f2mr2lre7e4">fatcat:32k2aqze7jbizg3f2mr2lre7e4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200907004452/https://arxiv.org/pdf/1912.13328v2.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/d1/c5/d1c5b0a7c08baa83f8ae900d3f90f60c80e639a2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1912.13328v2" 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>

Structure and Colour in Triangle-Free Graphs

N. R. Aravind, Stijn Cambie, Wouter Cames van Batenburg, Rémi De Joannis de Verclos, Ross J. Kang, Viresh Patel
<span title="2021-06-18">2021</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;
Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\  ...  and induced $H$-free graph has chromatic number at most $c D/\log D$.  ...  We are grateful to Alex Scott and two anonymous referees for helpful comments leading to improvements in the presentation of our results.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/9267">doi:10.37236/9267</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iid2rprsabgxzcj4jg2hsmk6hq">fatcat:iid2rprsabgxzcj4jg2hsmk6hq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210703001056/https://www.combinatorics.org/ojs/index.php/eljc/article/download/v28i2p47/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/cc/74/cc7430adb3651357aa718da93428e3c0c575ee70.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/9267"> <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>

Algorithms [chapter]

<span title="2011-10-28">2011</span> <i title="John Wiley &amp; Sons, Inc."> Graph Coloring Problems </i> &nbsp;
Graphs 122 7.2 Grunbaum's Girth Problem 123 7.3 Smallest Triangle-Free/t-Chromatic Graphs 123 7.4 Large Bipartite Subgraphs of Triangle-Free Graphs 126 7.5 Sparse Subgraphs 126 7.6 Number  ...  of Hamilton Cycles . 82 4.6 Brooks' Theorem for Triangle-Free Graphs 83 4.7 Graphs Without Large Complete Subgraphs 85 4.8 ^-Chromatic Graphs of Maximum Degree k 85 4.9 Total Coloring 86  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/9781118032497.ch10">doi:10.1002/9781118032497.ch10</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/374tktuvgvekni4fnz3dgbytjm">fatcat:374tktuvgvekni4fnz3dgbytjm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170814092450/http://www.gbv.de/dms/goettingen/152233997.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/85/c7/85c7cc15bc81c920cfd0f618ceca083713003efb.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/9781118032497.ch10"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

Triangle-Free Subgraphs of Random Graphs

Peter Allen, Julia Böttcher, Barnaby Roberts, Yoshiharu Kohayakawa
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fhi2xwpnh5gmlgof2idwu5wlgq" style="color: black;">Electronic Notes in Discrete Mathematics</a> </i> &nbsp;
The Andrásfai-Erdős-Sós Theorem [2] states that all triangle-free graphs on n vertices with minimum degree strictly greater than 2n/5 are bipartite.  ...  Thomassen [11] proved that when the minimum degree condition is relaxed to ( 1 3 + ε)n, the result is still guaranteed to be r ε -partite, where r ε does not depend on n.  ...  This motivates the question of which additional restrictions on the class of triangle-free graphs allow for a bound on the chromatic number.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2015.06.055">doi:10.1016/j.endm.2015.06.055</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wvqdi4j3knd7bfyrn7tkmgafoq">fatcat:wvqdi4j3knd7bfyrn7tkmgafoq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922050403/http://eprints.lse.ac.uk/64640/1/__lse.ac.uk_storage_LIBRARY_Secondary_libfile_shared_repository_Content_Allen%2C%20Peter_Triangle-free%20subgraphs_Allen_Triangle-free%20subgraphs_2015.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/78/7f/787f415d0982501727c37e40f41c3bb02932782f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2015.06.055"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum [article]

Csilla Bujtás, Zsolt Tuza
<span title="2015-08-07">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide k-colorability for every k > 2 (and remains intractable even for graphs of maximum degree 9 if k=3).  ...  In fact for every integer k> 3 there exists a K_3-WORM colorable graph in which the minimum number of colors is exactly k.  ...  Large W − and gap in the chromatic spectrumWe start with a connected triangle-free graph G k whose chromatic number is equal to k.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1508.01759v1">arXiv:1508.01759v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/atpv2yi5jfa2nlvbccvtiswkqq">fatcat:atpv2yi5jfa2nlvbccvtiswkqq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191025191244/https://arxiv.org/pdf/1508.01759v1.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/5b/4e/5b4e8dd021ffa0d47fe2cc9d4d9815d705d09763.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1508.01759v1" 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>

Fractional Coloring of Triangle-Free Planar Graphs

Zdeněk Dvořák, Jean-Sébastien Sereni, Jan Volec
<span title="2015-10-16">2015</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 prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-3/(3n+1)$.  ...  On the other hand, Jones's construction shows the existence of triangle-free planar graphs with fractional chromatic number arbitrarily close to 3.  ...  The purpose of this work is to answer this question. We do so by establishing the following upper bound on the fractional chromatic number of triangle-free planar n-vertex graphs, which depends on n.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/4135">doi:10.37236/4135</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hdqiupe6kjdnndm33mcjiv4eva">fatcat:hdqiupe6kjdnndm33mcjiv4eva</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200323222640/https://hal.archives-ouvertes.fr/hal-00950493/document" 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/4b/e6/4be64c67e4df6093b481caade45a3130550d9889.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/4135"> <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>

Separation Choosability and Dense Bipartite Induced Subgraphs

Louis Esperet, Ross Kang, Stéphan Thomassé
<span title="2019-02-26">2019</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/domxx2wewzae3e2m2ziqaqbyhm" style="color: black;">Combinatorics, probability &amp; computing</a> </i> &nbsp;
For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?  ...  We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon.  ...  confirmed Conjecture 1.3) and have nearly settled Conjecture 1.5 (and thus confirmed Conjecture 1.6), in that they have established ( log d/ log log d) bipartite induced minimum degree in K r -free graphs  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548319000026">doi:10.1017/s0963548319000026</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yjnunsgmvvcstlci77s3dou4hy">fatcat:yjnunsgmvvcstlci77s3dou4hy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200508065906/https://www.cambridge.org/core/services/aop-cambridge-core/content/view/265127482211C42CA56294C8468D0FEF/S0963548319000026a.pdf/div-class-title-separation-choosability-and-dense-bipartite-induced-subgraphs-div.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/a7/b5/a7b585d3cd82287b810d4dc7898e5f0fdb33ad6e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548319000026"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> cambridge.org </button> </a>

Triangle-Free Subgraphs of Random Graphs

PETER ALLEN, JULIA BÖTTCHER, YOSHIHARU KOHAYAKAWA, BARNABY ROBERTS
<span title="2017-08-14">2017</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/domxx2wewzae3e2m2ziqaqbyhm" style="color: black;">Combinatorics, probability &amp; computing</a> </i> &nbsp;
Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree.  ...  We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p −1 n)-close to bipartite, and each spanning triangle-free subgraph  ...  This motivates the question of which additional restrictions on the class of triangle-free graphs allow for a bound on the chromatic number.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548317000219">doi:10.1017/s0963548317000219</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cok2js2kyjb3xhp3d5n3pqquye">fatcat:cok2js2kyjb3xhp3d5n3pqquye</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190307074236/http://pdfs.semanticscholar.org/d33c/dc997756898472440b9f2db6e61cbc7af558.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/d3/3c/d33cdc997756898472440b9f2db6e61cbc7af558.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548317000219"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> cambridge.org </button> </a>
&laquo; Previous Showing results 1 &mdash; 15 out of 3,085 results