Before revision:  Let G=(V,E) be a vertex-colored interval graph with a k-coloring \gamma \colon V \rightarrow \{1,\ldots,k\} for some k \in \mathbb N . A subset of vertices S\subseteq V is called f-balanced if S contains f vertices from each color class.
After revision:  Let G=(V,E) be a vertex-colored interval graph with a k-coloring \gamma \colon V \rightarrow \{1,\ldots,k\} that maps all vertices in G onto k colors . A subset of vertices S\subseteq V is called f-balanced if S contains f vertices from each color class.