Star-Critical Gallai–Ramsey Numbers of Graphs

The Gallai–Ramsey number $$gr_{k}(K_{3}: H_{1}, H_{2}, \cdots , H_{k})$$ is the smallest integer n such that every k-edge-colored $$K_{n}$$ contains either a rainbow $$K_3$$ or monochromatic $$H_{i}$$ in color i for some $$i\in [k]$$ . We define star-critical $$gr_{k}^{*}(K_3: as s $$K_{n}-K_{1, n-1-s}$$ When $$H=H_{1}=\cdots =H_{k}$$ we simply denote $$gr_{k}^{*}(K_{3}: by H)$$ determine numbers complete graphs and small graphs. Furthermore, show exponential k if H not bipartite, linear bipartite but star constant (not depending on k) star.