$k$-Distance Magic Labeling and Long Brush Graphs

We define a labeling $f:$ $V(G)$ $\rightarrow$ $\{1, 2, \ldots, n\}$ on a graph $G$ of order $n \geq 3$ as a \emph{$k$-distance magic} ($k$-DM) if $\sum_{w\in \partial N_k(u)}{ f(w)}$ is a constant and independent of $u\in V(G)$ where $\partial N_k(u)$ = $\{v\in V(G): d(u, v) = k\}$, $k\in\mathbb{N}$. Graph $G$ is called a \emph{$k$-DM} if it has a $k$-DM labeling(L). Long Brush is a graph $G$ with $V(G)$ = $\{u_1, u_2, . . . , u_n,$ $v_1, v_2, . . . , v_{m}\}$, a path $P_n$ = $u_1$ $u_2$ . . . $u_n$ and $E(G)$ = $E(P_n)$ $\cup$ $\{u_1v_i:$ $i$ = 1 to $m\}$ $\cup$ $E(<v_1, v_2, . . . , v_{m}>)$, $m+n \geq 3$ and $m,n\in\mathbb{N}$. We denoted this graph by $LP_{n, m}$. In this paper, using partition techniques, we obtain families of $k$-DM graphs and prove that $(i)$ For $k,n \geq 3$, $m \geq 2$ and $k,m,n\in\mathbb{N}$, $LP_{n,m}$ is $k$-DM if and only if $m(m-1) \leq 2n$ and $k$ = $n$; (ii) For every $k\in\mathbb{N}_0$ and a given $m \geq 2$, $LP_{\frac{m(m-1)}{2}+k, m}$ is a $(\frac{m(m-1)}{2}+k)$-DM graph; (iii) For $m \geq 3$, $LP_{1,m}$ = $K_1(u_1)+(K_{m_1} \cup K_{m_2} \cup ... \cup K_{m_x})$, $x \geq 2$, $1 \leq m_1 \leq m_2 \leq ... \leq m_x$, $m_1+m_2+...+m_x$ = $m$, $m_1+m_2 \geq 3$ and $m_1,m_2,...,m_x,x\in\mathbb{N}$, $LP_{1,m}$ is 2-DM if and only if $u_1$ is assigned with a suitable $j$ and $J_{m+1}\setminus \{j\}$ is partitioned into $x$ constant sum partites of orders $m_1,m_2,...,m_x$, $1 \leq j \leq m+1$; (iv) For $m \geq 2$ if $LP_{2,m}$ contains two pendant vertices, then $LP_{2,m}$ is not a $2$-DM graph; (v) For $m \geq 2$ and $n \geq 3$, if $LP_{n,m}$ contains three pendant vertices, then $LP_{n,m}$ is not a $2$-DM graph; and (vi) for $m_1$ = 1 to 22, we obtain all possible values of $m$ for which $LP_{1, m}$ = $u_1 + (K_{m_1} \cup K_{m_2})$ is 2-DM, $m_1 \leq m_2$, $m = m_1+m_2 \geq 3$ and $m_1,m_2\in\mathbb{N}$.