the upper domatic number of powers of graphs

let a and b be two disjoint subsets of the vertex set v of a graph g. the set a is said to dominate b, denoted by a→b, if for every vertex u∈b there exists a vertex v∈a such that uv∈e(g). for any graph g, a partition π={v1, v2, …, vp} of the vertex set v is an \textit{upper domatic partition} if vi→vj or vj→vi or both for every vi,vj∈π, whenever i≠j. the \textit{upper domatic number} d(g) is the maximum order of an upper domatic partition. in this paper, we study the upper domatic number of powers of graphs and examine the special case when power is 2. we also show that the upper domatic number of kth power of a graph can be viewed as its k-upper domatic number.