A graph G = ( V, E ) is uniquely colorable if the chromatic number χ ( G ) = n and every n - coloring of G induces the same partition of V. In this paper, we introduce a new kind of graph called γ - uniquely colorable graphs. We obtain a necessary and sufficient condition for a graph to be γ - uniquely colorable graphs. We provide a constructive characterization of γ - uniquely colorable trees. © Published under licence by IOP Publishing Ltd.