On set colorings of complete bipartite graphs

In European J. Combin. 30 (2009), 986-995, S. M. Hegde recently introduced set colorings of a graph G as an assignment (function) of distinct subsets of a finite set X of colors to the vertices of G, where the colors of the edges are obtained as the symmetric difference of the sets assigned to their end vertices (which are also distinct). A set coloring is called a proper set coloring if all the nonempty subsets of X are obtained on the edges. A graph is called properly set colorable if it admits a proper set coloring. In this paper we give a proof for Hegde's conjecture that the complete bipartite graph K-a,K-b is properly set colorable if and only if one of the partition sets is of cardinality 1, and the other one of cardinality 2(n) - 1 for some positive integer n.