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 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. R(G_1, G_2)= {|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 R(C_4,C_5)=19, R(C_6,C_6)=26, R(P_4, C_5) = 14, R(P_4, P_5) = 10, R(P_4, P_6) = 14, R(P_5, P_5) = 11, R(P_3, P_5)=7 and R(P_3, P_6) = 8.ize Ramsey numbers of paths and cycles