A note on adjacent vertex distinguishing colorings of graphs

For an assignment of numbers to the vertices of a graph, let S[u]S[u] be the sum of the labels of all the vertices in the closed neighborhood of uu, for a vertex uu. Such an assignment is called closed distinguishing   if S[u]≠S[v]S[u]≠S[v] for any two adjacent vertices uu and vv unless the closed neighborhoods of uu and vv coincide. In this note we investigate dis[G], the smallest integer kk such that there is a closed distinguishing labeling of GG using labels from {1,…,k}{1,…,k}. We prove that dis[G]≤Δ2−Δ+1, where ΔΔ is the maximum degree of GG. This result is sharp. We also consider a list-version of the function dis[G] and give a number of related results.