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Abstract : In [S. M. Hegde, Set colorings of graphs, European J. Combin. 30 (2009) 986-995.] Hegde introduced the notion of set colorings of a graph G as an assignment of distinct subsets of a finite set X of n colors to the vertices of G such that all the colors of the edges which are obtained as the symmetric differences of the subsets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of G are the set of all nonempty subsets of X, then the coloring is said to be a strong set-coloring and G is said to be strongly set-colorable. In this paper, we report some new necessary conditions and propose a conjuncture for the sufficient condition for a graph to admit strong set-coloring. We also identify and characterize some new classes of graphs admitting strong set-coloring. In addition to these, we also propose strategies to construct infinite families graphs admitting strong set-coloring. © 2019 World Scientific Publishing Company. Notes: cited By 0; Article in Press