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

Abstract

Given a graph $G$ and a positive integer $k$, define the \emph{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 obtain general upper and lower bounds on the Gallai-Ramsey numbers for the graph $G=S_t^r$ obtained from a star of order $t$ by adding $r$ extra independent edges between leaves of the star so there are $r$ triangles and $t-2r-1$ pendent edges in $S_t^r$. We also prove some sharp results when $t=2$.