Gallai–Ramsey Numbers Involving a Rainbow 4-Path

Given two non-empty graphs G, H and a positive integer k, the Gallai–Ramsey number grk(G:H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {gr}}_k(G:H)$$\end{document} is defined as the minimum integer N such that for all n≥N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge N$$\end{document}, every k-edge-coloring of Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n$$\end{document} contains either a rainbow colored copy of G or a monochromatic copy of H. In this paper, we got some exact values or bounds for grk(P5:H)(k≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {gr}}_k(P_5:H) \ (k\ge 3)$$\end{document} if H is a general graph or a star with extra independent edges or a pineapple.