Gallai–Ramsey Numbers for a Class of Graphs with Five Vertices

Abstract

Given two graphs G and H, the k-colored Gallai–Ramsey number $$gr_k(G : H)$$ is defined to be the minimum integer n such that every k-coloring of the complete graph on n vertices contains either a rainbow copy of G or a monochromatic copy of H. In this paper, we consider $$gr_k(K_3 : H)$$ , where H is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai–Ramsey numbers for eight of them have been studied step by step in several papers. We determine all the Gallai–Ramsey numbers for the remaining five graphs, and we also obtain some related results for a class of unicyclic graphs. As applications, we find the mixed Ramsey spectra $$S(n; H, K_3)$$ for these graphs by using the Gallai–Ramsey numbers.