Some results on set colorings of directed trees

Abstract

A set coloring of the digraph D is an assignment (function) of distinct subsets of a finite set X of colors to the vertices of the digraph, where the color of an arc, say (u, v) is obtained by applying the set difference from the set assigned to the vertex v to the set assigned to the vertex u which arc also distinct. A set coloring is called a strong set coloring if sets on the vertices and arcs are distinct and together form the set of all non empty subsets of X. A set coloring is called a proper set coloring if all the non empty subsets of X are obtained on the arcs of D. A digraph is called a strongly set colorable (properly set colorable) if it admits a strong set coloring (proper set coloring).

In this paper we find some classes of directed trees which admit a strong set coloring and construction of strongly set colorable directed tree Tn.