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.