k-Product Cordial Labeling of Path Graphs

In 2012, Ponraj et al. defined a concept of k-product cordial labeling as follows: Let f be a map from $V\left(G\right)$ to $\left\{\mathrm{0,1,}？\mathrm{,}k？1\right\}$ where k is an integer, $1\le k\le \left|V\left(G\right)\right|$ . For each edge $uv$ assign the label $f\left(u\right)f\left(v\right)\left(\mathrm{mod}k\right)$ . f is called a k-product cordial labeling if $\left|v_f\left(i\right)？v_f\left(j\right)\right|\le 1$ , and $\left|e_f\left(i\right)？e_f\left(j\right)\right|\le 1$ , $i\mathrm{,}j\in \left\{\mathrm{0,1,}？\mathrm{,}k？1\right\}$ , where $v_f\left(x\right)$ and