Exact Values for Some Size Ramsey Numbers of Paths and Cycles - Li, Xiangmei; Fahad, Asfand; Zhou, Xiaoqing; Yang, Hong (Frontiers in Physics, 2020)
Exact values for some sFor graphs $G_1$, $G_2$ and $G$, if every 2-coloring (red and blue) of the edges of $G$ results in either a copy of blue $G_1$ or a copy of red $G_2$, we write $G → (G_1, G_2)$. 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 $G → (G_1, G_2)$, i.e. $\hat{R}(G_1, G_2)= \min\{|E(G)|: G → (G_1, G_2)\}$. 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