The chromatic number $\chi(G)$ of a graph $G$ is the smallest number of colors needed to color the vertices of $G$ so that no two adjacent vertices share the same color.
I'm not certain if the following problem has been researched before.
Problem 1. Given $k$ different colors, what is the maximum number of vertices in a graph that one can color using only these $k$ colors?
I recall this problem coming from the previous link. Graph $G$ is the skeleton of the pseudo-rhombicuboctahedron with every square face replaced by a clique. Using 4 colors isn't sufficient to color all the vertices of $G$ （thanks to Eric Nathan Stucky). However, what's interesting is, with these 4 colors, what is the maximum number of vertices of $G$ that can be colored? Of course, if an algorithm can be developed, it would be even more helpful for analyzing certain graphs.
I've abstracted it, though. I believe this problem might have been studied by researchers in the past.
Taking it further.
Problem 2. Color (fixed) $k$ vertices of $G$ with these $k$ colors first (ensuring adjacent vertices have distinct colors), is it possible to efficiently determine an upper bound on the number of vertices in $G$ that can be colored?
