Set colorings of digraphs

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 are 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. 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 give some necessary conditions for a digraph to admit a strong set coloring (proper set coloring), characterize strongly (proper) set colorable digraphs such as directed stars, directed bistars etc.