  Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G :
H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of
the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a
monochromatic copy of $H$. In this paper, we consider $gr_k(K_3 : H)$ where $H$
is a connected graph with five vertices and at most six edges. There are in
total thirteen graphs in this graph class, and the Gallai-Ramsey numbers for
some of them have been studied step by step in several papers. We determine all
the Gallai-Ramsey numbers for the remaining graphs, and we also obtain some
related results for a class of unicyclic graphs.
