Conditional and Unique Coloring of Graphs

For integers k > 0 and 0 < r ≤ � (where r ≤ k), a conditional (k,r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex v of degree d(v) in G is adjacent to vertices with at least min{r,d(v)} differently colored neighbors. The smallest integer k for which a graph G has a conditional (k,r)-coloring is