The Construction of 3-TEPC labeling for Stellation of octagonal grid graph

Resumo

Consider $G$ is a finite and simple graph with vertex set $V(G)$ and edge set $E(G)$. An edge labeling $\rho:E(G)\rightarrow\{0,1,...,r\}$, where $r$ is an integer, $2\leq r\leq|E(G)|$, induces a vertex labeling $\rho^{*}:V(G)\rightarrow\{0,1,...,r-1\}$, defined in such a way that $\rho^{*}(v)= {\prod_{i=1}^{n}\rho(e_{i})}$ (mod r), where $e_{i}$ are the edges incident to the vertex $v$. The mapping $\rho$ is called an $r$-total edge product cordial labeling, r-TEPC labeling, of $G$ if $\mid e_{\rho}(i)+v_{\rho^{*}}(i)-e_{\rho}(j)+v_{\rho^{*}}(j)\mid\leq1$ for every $0\leq i , j \leq r-1$, where the numbers of edges and vertices labeled with integer $i$ are denoted by $e_{\rho}(i)$ and $v_{\rho^{*}}(i)$ respectively. In this paper we have shown that the stellation of an octagonal grid graph admits a 3-TEPC labeling.