Ramsey and Gallai-Ramsey Numbers of Cycles and Books

Given two non-empty graphs G, H and a positive integer k, the Gallai-Ramsey number grk(G: H) is defined as the minimum integer N such that for all n ≥ N, every exact k-edge-coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H. Denote grk′(G: H) as the minimum integer N such that for all n ≥ N, every edge-coloring of Kn using at most k colors contains either a rainbow copy of G or a monochromatic copy of H. In this paper, we get some exact values or bounds for grk(P5: H) and grk′(P5: H), where H is a cycle or a book graph. In addition, our results support a conjecture of Li, Besse, Magnant, Wang and Watts in 2020.