The profile of the Cartesian product of graphs

Given a graph GG, a proper labeling  ff of GG is a one-to-one function from V(G)V(G) onto {1,2,…,|V(G)|}{1,2,…,|V(G)|}. For a proper labeling ff of GG, the profile width  wf(v)wf(v) of a vertex vv is the minimum value of f(v)−f(x)f(v)−f(x), where xx belongs to the closed neighborhood of vv. The profile of a proper labeling  ffof  GG, denoted by Pf(G)Pf(G), is the sum of all the wf(v)wf(v), where v∈V(G)v∈V(G). The profile of  GG is the minimum value of Pf(G)Pf(G), where ff runs over all proper labeling of GG. In this paper, we show that if the vertices of a graph GG can be ordered to satisfy a special neighborhood property, then so can the graph G×QnG×Qn. This can be used to determine the profile of QnQn and Km×QnKm×Qn.