Ramsey and Gallai–Ramsey numbers for stars with extra independent edges

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 Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai–Ramsey numbers for the graph G=Str obtained from a star of order t by including r extra independent edges between leaves of the star so there are r triangles and t−2r−1 pendant edges in Str. We also prove some sharp results when r=2.