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A (proper) vertex coloring of a graph is a labelling of the graph’s vertices with colors such that no two vertices sharing the same edge have the same color. A coloring using at most $k$ colors is called a (proper) $k$-coloring.

The $k$-coloring partitions the vertex set of a graph into $k$ sets. What do I say to refer to one of these sets of vertices? For example, "fix a vertex coloring of $G$ and let $A$ be the (blank) of largest order."

Is there a standard terminology? Also, if there are multiple terms, what other words for these might I see floating around?

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The most common term for the collection of vertices receiving the color $i$ is "the $i^\text{th}$ color class" (or just "color class" if you do not wish to specify its color).

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