A Note on \(3\)-Equitable Labelings Of Multiple Shells

Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A vertex labelling \( f: V \rightarrow \{0, 1, 2\} \) induces an edge labelling \( \overline{f}: E \rightarrow \{0, 1, 2\} \) 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 \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \) for \( 0 \leq i, j \leq 2 \). In this paper, we show that every multiple shell \( MS\{n_1^{t_1}, \ldots, n_r^{t_r}\} \) is 3-equitable for all positive integers \( n_1, \ldots, n_r, t_1, \ldots, t_r \).