Gallai–Ramsey Numbers Involving a Rainbow 4-Path

Given two non-empty graphs G, H and a positive integer k, the Gallai–Ramsey number $${\text {gr}}_k(G:H)$$ is defined as minimum N such that for all $$n\ge N$$ , every k-edge-coloring of $$K_n$$ contains either rainbow colored copy G or monochromatic H. In this paper, we got some exact values bounds {gr}}_k(P_5:H) \ (k\ge 3)$$ if general graph star with extra independent edges pineapple.