Malaysian Journal of Mathematical Sciences, January 2022, Vol. 16, No. 1

Edge Irregular Reflexive Labeling for Some Classes of Plane Graphs

Yoong, K. K., Hasni, R., Lau, G. C., and Irfan, M.

Corresponding Email: hroslan@umt.edu.my

Received date: 17 June 2021 Accepted date: 14 October 2021

Abstract: For a graph $G$, we define a total $k$-labeling $\varphi$ as a combination of an edge labeling $\varphi_e(x)\to\{1,\,2,\,\ldots,\,k_e\}$ and a vertex labeling $\varphi_v(x) \to \{0,\,2,\,\ldots,\,2k_v\}$, such that $\varphi(x)= \varphi_v(x)$ if $x\in V(G)$ and $\varphi(x)= \varphi_e(x)$ if $x\in E(G)$, where $k=\,\mbox{max}\,\{k_e,\,2k_v\}$. The total $k$-labeling $\varphi$ is called an \(\textit{edge irregular reflexive $k$-labeling}\) of $G$, if for every two edges $xy$, $x^\prime y^\prime$ of $G$, one has $wt(xy)\neq wt(x^\prime y^\prime)$, where $wt(xy)=\varphi_v(x)+ \varphi_e(xy)+ \varphi_v(y)$. The smallest value of $k$ for which such labeling exists is called a \(\textit{reflexive edge strength}\) of $G$. In this paper, we study the edge irregular reflexive labeling on plane graphs and determine its reflexive edge strength.