Restricted size Ramsey number for P3 versus cycle

• Main Authors: Joanna Cyman, Tomasz Dzido
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2020-10-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: restricted size ramsey numbers, path, cycle
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/822

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>^*(<em>G,H</em>) is defined as <em>r</em>^*(<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>^*(<em>P</em>₃,<em>C_n</em>) for 7 ≤ <em>n</em> ≤ 12. We also give new upper bound <em>r</em>^*(<em>P</em>₃,<em>C</em>_n) ≤ 2<em>n</em>-2 for <em>n</em> ≥ 10 and <em>n</em> is even.