Gallai-Ramsey numbers for the union of stars

Abstract

For two graphs $G$, $H$ and a positive integer $k$, the Gallai--Ramsey number $gr_{k}(G:H)$ is defined as the minimum integer $n$ such that any $k$-edge-coloring of $K_{n}$ must contain either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we consider the exact values of $gr_{k}(G:K_{1,n}\cup K_{1,m})$, where $G\in \{P_{5}, K_{1,3}, P^{+}_{4}\}$ when $k\geq4$ and $P^{+}_{4}$ is the graph obtained from $P_{4}$ by adding an extra edge incident with an inner vertex.