Uniquely proper distinguishing colorable graphs
Kazem Khashyarmanesh (khashyar@ipm.ir)
A proper vertex-coloring of a graph G is called distinguishing, if the only automorphism preserving the colors is the identity. The minimum number of colors for such a coloring is denoted by χ D (G) . We say a graph G is uniquely proper distinguishing colorable (UPDC, for short) if and only if there exists only one partition of V(G) into χ D (G) independent sets such that the identity is the only automorphism of G preserving the partition. In this paper, we study the UPDC graphs. We show that a