Exact Values for Some Size Ramsey Numbers of Paths and Cycles - Li, Xiangmei; Fahad, Asfand; Zhou, Xiaoqing; Yang, Hong (Frontiers in Physics, 2020)
For the graphs G1, G2, and G, if every 2-coloring (red and blue) of the edges of G results in either a copy of blueG1 or a copy of redG2, we write G → (G1, G2). The size Ramsey number R^(G1,G2) is the smallest number e such that there is a graph G with size e satisfying G → (G1, G2), i.e., R^(G1,G2)=min{|E(G)|:G→(G1,G2)}. In this paper, by developing the procedure and algorithm, we determine exact values of the size Ramsey numbers of some paths and cycles. More precisely, we obtain that R^(C4,C5)=19, R^(C6,C6)=26, R^(P4,C5)=14, R^(P4,P5)=10, R^(P4,P6)=14, R^(P5,P5)=11, R^(P3,P5)=7 and R^(P3,P6)=8.