Vertex equitable labeling for ladder and snake related graphs

Let \(G\) be a graph with p vertices and q edges and \(A = {0, 1, 2, · · · ,\frac{q}{2}}\). A vertex labeling \(f : V (G) → A\) induces an edge labeling \(f^∗\) defined by \(f^∗ (uv) = f(u) + f(v)\) for all edges \(uv\). For \(a ∈ A\), let \(v_f (a)\) be the number of vertices \(v\) with\( f(v) = a\). A graph \(G\) is said to be vertex equitable if there exists a vertex labeling f such that for all \(a\) and \(b\) in A, \(|v_f (a) − v_f (b)| ≤ 1\) and the induced edge labels are \({1, 2, 3, · · · , q}\). In this paper, we prove that triangular ladder \(T L_n, L_n ⊙ mK_1, Q_n ⊙ K_1, T L_n⊙K_1\) and alternate triangular snake \(A(T_n)\) are vertex equitable graphs.