AUTHOR=Li Xiangmei , Fahad Asfand , Zhou Xiaoqing , Yang Hong 

TITLE=Exact Values for Some Size Ramsey Numbers of Paths and Cycles

JOURNAL=Frontiers in Physics

VOLUME=Volume 8 - 2020

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.00350

DOI=10.3389/fphy.2020.00350

ISSN=2296-424X

ABSTRACT=Exact values for some sFor graphs $G_1$, $G_2$ and $G$, if every 2-coloring (\emph{red} and \emph{blue}) of the edges of $G$ results in either a copy of \emph{blue} $G_1$ or a copy of \emph{red} $G_2$, we write $G \rightarrow (G_1, G_2)$. The size Ramsey number $\hat{R}(G_1, G_2)$ is the smallest number $e$ such that there is a graph $G$ with size $e$ satisfying $G \rightarrow (G_1, G_2)$, i.e. $\hat{R}(G_1, G_2)= \min\{|E(G)|: G \rightarrow (G_1, G_2)\}$. In this paper, by developing the procedure and algorithm, we determine some exact values of the size Ramsey number of paths and cycles. More precisely, we obtain that $\hat{R}(C_4,C_5)=19$, $\hat{R}(C_6,C_6)=26$, $\hat{R}(P_4, C_5) = 14$, $\hat{R}(P_4, P_5) = 10$, $\hat{R}(P_4, P_6) = 14$, $\hat{R}(P_5, P_5) = 11$, $\hat{R}(P_3, P_5)=7$ and $\hat{R}(P_3, P_6) = 8$.ize Ramsey numbers of paths and cycles