A Note on $3$-Equitable Labelings Of Multiple Shells

Abstract

Let $G$ be a graph with vertex set $V$ and edge set $E$. A vertex labelling $f:V\to \{0,1,2\}$ induces an edge labelling $\overline{f}:E\to \{0,1,2\}$ defined by $\overline{f}(uv)=|f(u)–f(v)|$. Let $v_f(0),v_f(1),v_f(2)$ denote the number of vertices $v$ with $f(v)=0,f(v)=1$ and $f(v)=2$ respectively. Let $e_f(0),e_f(1),e_f(2)$ be similarly defined. A graph is said to be 3-equitable if there exists a vertex labelling $f$ such that $|v_f(i)–v_f(j)|\le 1$ and $|e_f(i)–e_f(j)|\le 1$ for $0\le i,j\le 2$. In this paper, we show that every multiple shell $MS\{n_1^{t_1},\dots ,n_r^{t_r}\}$ is 3-equitable for all positive integers $n_1,\dots ,n_r,t_1,\dots ,t_r$.