Ramsey and Gallai-Ramsey Numbers for Two Classes of Unicyclic Graphs

Abstract

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of $$K_n$$ contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the 2-color Ramsey numbers for these two classes of graphs and use these to obtain general upper and lower bounds on the Gallai-Ramsey numbers.