Modular colorings of join of two special graphs

For k ≥ 2, a modular k-coloring of a graph G without isolated vertices is a coloring of the vertices of G with the elements in Zk having the property that for every two adjacent vertices of G, the sums of the colors of their neighbors are different in Zk . The minimum k for which G has a modular k- coloring is the modular chromatic number of G. In this paper, we determine the modular chromatic number of join of two special graphs.