A note on adjacent vertex distinguishing colorings number of graphs

For an assignment of numbers to the vertices of a graph, let S[u] be the sum of the labels of all the vertices in the closed neighborhood of u, for a vertex u. Such an assignment is called closed distinguishing if S[u] 6= S[v] for any two adjacent vertices u and v unless the closed neighborhoods of u and v coincide. In this note we investigate dis[G], the smallest integer k such that there is a closed distinguishing labeling of G using labels from {1, . . . , k}. We prove that dis[G] ≤ ∆2 − ∆ + 1, where ∆ is the maximum degree of G. This result is sharp. We also consider a list-version of the function dis[G] and give a number of related results.