On the Ramsey number of 4-cycle versus wheel

Abstract

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>For any fixed graphs $</span><span>G$ </span><span>and $</span><span>H$, </span><span>the Ramsey number $</span><span>R</span><span>(</span><span>G,H</span><span>)$ is the smallest positive integer $</span><span>n$ </span><span>such that for every graph $</span><span>F$ </span><span>on $</span><span>n$ </span><span>vertices must contain $</span><span>G$ </span><span>or the complement of $</span><span>F$ </span><span>contains $</span><span>H$. </span><span>The girth of graph $</span><span>G$ </span><span>is a length of the shortest cycle. A $</span><span>k$</span><span>-regular graph with the girth $</span><span>g$ </span><span>is called a $(</span><span>k,g</span><span>)$-graph. If the number of of vertices in $(</span><span>k,g</span><span>)$-graph is minimized then we call this graph a $(</span><span>k,g</span><span>)$-cage. In this paper, we derive the bounds of Ramsey number $</span><span>R</span><span>(</span><span>C_</span><span>4</span><span>,W_</span><span>n</span><span>)$ for some values of $</span><span>n$</span><span>. By modifying $(</span><span>k, </span><span>5)$-graphs, for $</span><span>k </span><span>= 7$ or $9$, we construct these corresponding $(</span><span>C_</span><span>4</span><span>,W_</span><span>n</span><span>)$-good graphs. </span></p></div></div></div>