Locally Restricted Colorings of Graphs

Let G be a simple graph and f a function from the vertices of G to the set of positive integers. An (f, n)-coloring of G is an assignment of n colors to the vertices of G such that each vertex x is adjacent to less than f(x) vertices with the same color as x. The minimum n such that an (f, n)-coloring of G exists is defined to be the f chromatic number of G. In this paper, we address a study of this kind of locally restricted coloring.