On some Gallai Ramsey numbers

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallaik-coloring is a Gallai coloring that useskcolors. Given a graphHand an integerk≥ 1, the Gallai-Ramsey numberGR_k(H) ofHis the least positive integerNsuch that every Gallaik-coloring of the complete graphK_Ncontains a monochromatic copy ofH. In this report, we will talk about some new results and problems on the Gallai-Ramsey numbers for cycles, complete graphs and complete bipartite graphs.