Gallai–Ramsey Numbers of Odd Cycles and Complete Bipartite Graphs

Abstract

For graphs G and H and integer $$k\ge 1$$ , the Gallai–Ramsey number $$gr_k(G:H)$$ is defined to be the minimum integer N such that if $$K_N$$ is edge-colored with k colors, then there is either a rainbow G or a monochromatic H. It is known that $$gr_k(K_3:C_{2n+1})$$ is exponential in k. In this note, we improve the upper bound for $$gr_k(K_3:C_{2n+1})$$ obtained by Hall et al., and give bounds for $$gr_k(K_3:K_{m,n})$$ .