Restricted size Ramsey number for P3 versus cycle
Let <em>F</em>, <em>G</em> and <em>H</em> be simple graphs. We say <em>F</em> → (<em>G</em>,<em>H</em>) if for every 2-coloring of the edges of <em>F</em> there exists a red copy of <em>G</em> or a blue c...
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Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia
2020-10-01
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doaj-3d690244b1d94d81bb94918f77e385892021-03-11T01:13:06ZengIndonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), IndonesiaElectronic Journal of Graph Theory and Applications2338-22872020-10-018236537210.5614/ejgta.2020.8.2.12188Restricted size Ramsey number for P3 versus cycleJoanna Cyman0Tomasz Dzido1Gda\'nsk University of Technology, Department of Technical Physics and Applied Mathematics, Narutowicza 11/12, 80-952 Gda\'nsk, PolandInstitute of Informatics, University of Gda\'{n}sk, Wita Stwosza 57, 80-952 Gda\'{n}sk, PolandLet <em>F</em>, <em>G</em> and <em>H</em> be simple graphs. We say <em>F</em> → (<em>G</em>,<em>H</em>) if for every 2-coloring of the edges of <em>F</em> there exists a red copy of <em>G</em> or a blue copy of <em>H</em> in <em>F</em>. The Ramsey number <em>r</em>(<em>G,H</em>) is defined as <em>r</em>(<em>G,H</em>) = min{|<em>V</em>(<em>F</em>)|: <em>F</em> → (<em>G,H</em>)}, while the restricted size Ramsey number <em>r</em><sup>*</sup>(<em>G,H</em>) is defined as <em>r</em><sup>*</sup>(<em>G,H</em>) = min{|<em>E</em>(<em>F</em>)|: <em>F</em> → (<em>G,H</em>),|<em>V</em>(<em>F</em>)| = <em>r</em>(<em>G,H</em>)}. In this paper we determine previously unknown restricted size Ramsey numbers <em>r</em><sup>*</sup>(<em>P</em><sub>3</sub>,<em>C<sub>n</sub></em>) for 7 ≤ <em>n</em> ≤ 12. We also give new upper bound <em>r</em><sup>*</sup>(<em>P</em><sub>3</sub>,<em>C</em><sub>n</sub>) ≤ 2<em>n</em>-2 for <em>n</em> ≥ 10 and <em>n</em> is even.https://www.ejgta.org/index.php/ejgta/article/view/822restricted size ramsey numbers, path, cycle |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Joanna Cyman Tomasz Dzido |
spellingShingle |
Joanna Cyman Tomasz Dzido Restricted size Ramsey number for P3 versus cycle Electronic Journal of Graph Theory and Applications restricted size ramsey numbers, path, cycle |
author_facet |
Joanna Cyman Tomasz Dzido |
author_sort |
Joanna Cyman |
title |
Restricted size Ramsey number for P3 versus cycle |
title_short |
Restricted size Ramsey number for P3 versus cycle |
title_full |
Restricted size Ramsey number for P3 versus cycle |
title_fullStr |
Restricted size Ramsey number for P3 versus cycle |
title_full_unstemmed |
Restricted size Ramsey number for P3 versus cycle |
title_sort |
restricted size ramsey number for p3 versus cycle |
publisher |
Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia |
series |
Electronic Journal of Graph Theory and Applications |
issn |
2338-2287 |
publishDate |
2020-10-01 |
description |
Let <em>F</em>, <em>G</em> and <em>H</em> be simple graphs. We say <em>F</em> → (<em>G</em>,<em>H</em>) if for every 2-coloring of the edges of <em>F</em> there exists a red copy of <em>G</em> or a blue copy of <em>H</em> in <em>F</em>. The Ramsey number <em>r</em>(<em>G,H</em>) is defined as <em>r</em>(<em>G,H</em>) = min{|<em>V</em>(<em>F</em>)|: <em>F</em> → (<em>G,H</em>)}, while the restricted size Ramsey number <em>r</em><sup>*</sup>(<em>G,H</em>) is defined as <em>r</em><sup>*</sup>(<em>G,H</em>) = min{|<em>E</em>(<em>F</em>)|: <em>F</em> → (<em>G,H</em>),|<em>V</em>(<em>F</em>)| = <em>r</em>(<em>G,H</em>)}. In this paper we determine previously unknown restricted size Ramsey numbers <em>r</em><sup>*</sup>(<em>P</em><sub>3</sub>,<em>C<sub>n</sub></em>) for 7 ≤ <em>n</em> ≤ 12. We also give new upper bound <em>r</em><sup>*</sup>(<em>P</em><sub>3</sub>,<em>C</em><sub>n</sub>) ≤ 2<em>n</em>-2 for <em>n</em> ≥ 10 and <em>n</em> is even. |
topic |
restricted size ramsey numbers, path, cycle |
url |
https://www.ejgta.org/index.php/ejgta/article/view/822 |
work_keys_str_mv |
AT joannacyman restrictedsizeramseynumberforp3versuscycle AT tomaszdzido restrictedsizeramseynumberforp3versuscycle |
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1714790728812462080 |