EDGE-COLORING VERTEX-WEIGHTING OF SOME PRODUCT GRAPHS

Let $G$ be a graph. A $k-$vertex weighting of a graph $G$ is a mapping $w:V(G)\rightarrow\{1,2,3,\dots,k\}.$ A $k-$vertex weighting induces an edge labeling $f_w:E(G)\rightarrow\mathbb{N}$ such that $f_w(uv)=w(u)+w(v).$ Such a labeling is called an edge-coloring $k-$weighting if $f_w(e)\ne f_w(e')$ for any two adjacent edges $e$ and $e'.$ Denote by $\mu'(G)$ the minimum $k$ for $G$ to admit an edge-coloring $k-$vertex weighting. In this paper, we determine $\mu'(G)$ for some product graphs.