Conditional colorings of graphs

For an integer r > 0, a conditional (k, r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex of degree at least r in G will be adjacent to vertices with at least r different colors. The smallest integer k for which a graph G has a conditional k-coloring is the r-conditional chromatic number χr(G). In this paper, the behavior and bounds of conditional chromatic number of a graph G and its generalization are investigated.