Incidence coloring of Cartesian product graphs

For a vertex v∈V(G), the incidence neighborhood of v, denoted by IN(v), is the set {(v,e):e∈E(G) and v is incident with e}∪{(u,e):e=vu∈E(G)}. Let Sσ(v) denote the set of colors assigned to IN(v) in G under incidence coloring σ and s(σ)=max{|Sσ(v)|:v∈V(G)}. Let G1□G2 denote the Cartesian product of graphs G1 and G2. Let σi be an incidence coloring of graph Gi and n(σi) the number of colors used by σi for i∈{1,2}.