Cylindrical Grid Graphs $P_m \Box C_n$ are Non-Distance Magic

Abstract: A bijective mapping $f: V(G) \rightarrow \left{1,2,\ldots,n\right}$ is called a \emph{Distance Magic Labeling (DML) of $G$} if ~ ${\sum_{v \in N(u)}} f(v) $ is a constant for all $u\in V(G)$ where $G$ is a simple graph of order $n$ and $N(u)$ = ${v\in V(G):$ $uv\in E(G)}$. Graph $G$ is called a \emph{Distance Magic Graph (DMG)} if it has a DML, otherwise it is called a \emph{Non-Distance Magic (NDM) graph}. In 1996, Vilfred proposed a conjecture that cylindrical grid graphs $P_m \Box C_n$ are NDM for $m \geq 2$, $n \geq 3$ and $m,n\in\mathbb{N}$. Recently, the authors could prove the conjecture for the case when $m$ is even by introducing neighbourhood chains of Type-1 (NC-T1) and Type-2 (NC-T2). In this paper, they introduce neighbourhood chains of Type-3 (NC-T3) and using them completely settle the conjecture and also identify families of NDM graphs.