Vertex equitable labeling of firecracker and A-star graph Available to Purchase

Let G be a graph, V(G), E(G), and |E(G)| = q be its set of vertices, set of edges, and number of edges, respectively, and $Y={0,1,...,[q2]}$⁠. A vertex equitable labeling of G is a vertex labeling f: V(G)→Y, such that if y, z ∈ 2 Y, then |v_f(y)−v_f(z)| ≤ 1, where v_f(y) is the number of vertices x ∈ V(G) with f(x) = y for y ∈ Y, and f induces a bijective labeling f∗: E(G) → {1, 2, 3, …, q} defined by f∗(uv) = f(u) + f(v). If there exists a vertex equitable labeling of G, then G is said to be vertex equitable. Let n and k be positive integers, k ≥ 2. A firecracker graph F(n, k) is a graph constructed from n copies of star S_k by linking one leaf from each star. An A-star graph A(S_k) is a graph obtained from a shaped letter A graph and two copies of star S_k where each vertex of degree one on the shaped letter A graph is identified with the center of one copy of star S_k. We find that the fireccracker F(n, k), for even n, and A-star A(S_k) are vertex equitable.