The Full m Index Sets of P2×Pn

Abstract

Shiu and Kwong (2008) studied the full friendly index set of $P_2\times P_n$, which only addressed the cases where $m=0$ or 1. In this paper, we significantly extend their work by determining the full $m$ index set $M\left(P_2\times P_n\right)$ for all values of $m$. Our key approach is to utilize graph embedding and recursion methods to deduce $M\left(P_2\times P_n\right)$ for general $m$. In particular, we embed small graphs like $C_4$ and $K_2$ into $P_2\times P_n$ and apply recursive techniques to prove the main results. This work expands the scope of previous graph labeling studies and provides new insights into determining the full $m$ index set of product graphs. Given the broad range of applications for labeled graphs, this research can potentially impact fields like coding theory, communication network design, and more.