Finding Non-Distance Magic Graphs using neighbourhood chains

Abstract

Let $G$ be a graph of order $n$ and $N = \{N(u_{i})\}^k_{i=1}$ be a sequence of neighbourhood(nbh)s in $G$ where $N(u)$ = $\{v\in V(G):$ $uv\in E(G)\}$. \emph{Nbh sequence graph $H$ of} $N$ in $G$ is defined as the union of all induced subgraphs of closed nbh $N[u_{i}]$ in $G$, $1 \leq i \leq k$, $k\in\mathbb{N}$. A labeling $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 every $u\in V(G)$. $G$ 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 this paper, we define nbh walk, nbh trial, nbh path or nbh chain, nbh cycle, nbh sequence graph and nbh chains of Type-1 (NC-T1) and Type-2 (NC-T2). NC-T2 is formed on two NC-T1 of same length. We prove that (i) for $k \geq 2$ and $n \geq 3$, cylindrical grid graph $P_{k} \Box C_{n}$ contains NC-T2, $k,n \in \mathbb{N}$; (ii) graph containing NC-T1 of even length is NDM and (iii) partially settle a conjecture that graphs $P_m \Box C_n$ are NDM when $n$ is even, $m \geq 2$, $n \geq 3$ and $m,n\in\mathbb{N}$.