  A colored complete graph is said to be Gallai-colored if it contains no
rainbow triangle. This property has been shown to be equivalent to the
existence of a partition of the vertices (of every induced subgraph) in which
at most two colors appear on edges between the parts and at most one color
appears on edges in between each pair of parts. We extend this notion by
defining a coloring of a complete graph to be $k$-Gallai if every induced
subgraph has a nontrivial partition of the vertices such that there are at most
$k$ colors present in between parts of the partition. The generalized $(k,
\ell)$ Gallai-Ramsey number of a graph $H$ is then defined to be the minimum
number of vertices $N$ such that every $k$-Gallai coloring of a complete graph
$K_{n}$ with $n \geq N$ using at most $\ell$ colors contains a monochromatic
copy of $H$. We prove bounds on these generalized $(k, \ell)$ Gallai-Ramsey
numbers based on the structure of $H$, extending recent results for Gallai
colorings.
