There is a version of graph colouring where instead of insisting that each colour class is an independent set, you require each colour class to be a $c$-independent set. This is known as a colouring with clustering $c$. The clustered chromatic number of a class of graphs is the minimum $k$ such that for some integer $c$, every graph in the class is $k$-colourable with clustering $c$. Clustered chromatic number has received quite a lot of attention recently. For example, van den Heuvel and Wood proved that every $K_t$-minor free graph is $(2t-2)$-colourable with clustering at most $\lceil\frac{1}{2}(t-2) \rceil$. Note that the analous statement for the chromatic number is the linear Hadwidger's conjecture, and is still open.

See this survey by David Wood for more information. The survey also discusses colourings where each colour class has bounded maximum degree. These are known as defective colourings.

There is a version of graph colouring where instead of insisting that each colour class is an independent set, you require each colour class to be a $c$-independent set. This is known as a colouring with clustering $c$. The clustered chromatic number of a class of graphs is the minimum $k$ such that for some integer $c$, every graph in the class is $k$-colourable with clustering $c$. Clustered chromatic number has received quite a lot of attention recently. For example, van den Heuvel and Wood proved that every $K_t$-minor free graph is $(2t-2)$-colourable with clustering at most $\lceil\frac{1}{2}(t-2) \rceil$. Note that the analogous statement for the chromatic number is the linear Hadwidger's conjecture, and is still open.

See this survey by David Wood for more information. The survey also discusses colourings where each colour class has bounded maximum degree. These are known as defective colourings.

There is a version of the chromatic number where instead of insisting that each colour class is an independent set, you require each colour class to be a $c$-independent set. This is known as a colouring with clustering $c$. The clustered chromatic number of a class of graphs is the minimum $k$ such that for some integer $c$, every graph in the class is $k$-colourable with clustering $c$. Clustered chromatic number has received quite a lot of attention recently. For example, van den Heuvel and Wood proved that every $K_t$-minor free graph is $(2t-2)$-colourable with clustering at most $\lceil\frac{1}{2}(t-2) \rceil$. Note that the analous statement for the chromatic number is the linear Hadwidger's conjecture, and is still open.

See this survey by David Wood for more information. The survey also discusses colourings where each colour class has bounded maximum degree. These are known as defective colourings.

There is a version of graph colouring where instead of insisting that each colour class is an independent set, you require each colour class to be a $c$-independent set. This is known as a colouring with clustering $c$. The clustered chromatic number of a class of graphs is the minimum $k$ such that for some integer $c$, every graph in the class is $k$-colourable with clustering $c$. Clustered chromatic number has received quite a lot of attention recently. For example, van den Heuvel and Wood proved that every $K_t$-minor free graph is $(2t-2)$-colourable with clustering at most $\lceil\frac{1}{2}(t-2) \rceil$. Note that the analous statement for the chromatic number is the linear Hadwidger's conjecture, and is still open.

See this survey by David Wood for more information. The survey also discusses colourings where each colour class has bounded maximum degree. These are known as defective colourings.