The profile of the Cartesian product of graphs
Given a graph G G , a proper labeling f f of G G is a one-to-one function from V(G) V ( G ) onto {1,2,,|V(G)|} { 1 , 2 ,  , | V ( G ) | } . For a proper labeling f f of G G , the profile width wf(v) w f ( v ) of a vertex v v is the minimum value of f(v)?f(x) f ( v ) ? f ( x ) , where x x belongs to the closed neighborhood of v v . The profile of a proper labeling f f of G G , denoted by Pf(G) P f ( G ) , is the sum of all the wf(v) w f ( v ) , where v?V(G) v ? V ( G ) . The profile of G G is the minimum value of Pf(G) P f ( G ) , where f f runs over all proper labeling of G G . In this paper, we show that if the vertices of a graph G G can be ordered to satisfy a special neighborhood property, then so can the graph GQn G  Q n . This can be used to determine the profile of Qn Q n and KmQn K m  Q n .
profile cartesian product n n -cube n n n n n optimal labeling pmnl-graph p-property