Full friendly index sets of P2×PnP2×Pn

Let G=(V,E)G=(V,E) be a graph, a vertex labeling f:V→Z2f:V→Z2 induces an edge labeling f*:E→Z2f*:E→Z2 defined by f*(xy)=f(x)+f(y)f*(xy)=f(x)+f(y) for each xy∈Exy∈E. For each i∈Z2i∈Z2, define vf(i)=|f-1(i)|vf(i)=|f-1(i)| and ef(i)=|f*-1(i)|ef(i)=|f*-1(i)|. We call f   friendly if |vf(1)-vf(0)|⩽1|vf(1)-vf(0)|⩽1. The full friendly index set of G   is the set of all possible values of ef(1)-ef(0)ef(1)-ef(0), where f   is friendly. In this note, we study the full friendly index set of the grid graph P2×PnP2×Pn.