L(2, 1)-labelings of subdivisions of graphs

Given a graph G and a function h from E(G) to double-struck N, the h-subdivision of G, denoted by G_(h), is the graph obtained from G by replacing each edge uv in G with a path P : ux_uv¹x_uv² . . . x_uv^n-1 v, where n = h(uv). When h(e) = c is a constant for all e ∈ E(G), we use G_(c) to replace G_(h). Given a graph G, an L(2, 1)-labeling of G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x) - f(y)| ≥ 2 if d_G(x, y) = 1, and |f(x) - f(y)| ≥ 1 if d_G(x, y) = 2. A k-L(2, 1)-labeling is an L(2, 1)-labeling such that no label is greater than k. The L(2, 1)-labeling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2, 1)-labeling. We study the L(2, 1)-labeling numbers of subdivisions of graphs in this paper. We prove that λ(G₍₃₎) = Δ(G) + 1 for any graph G with Δ(G) ≥ 4, and show that λ(G_(h)) = Δ(G) + 1 if Δ(G) ≥ 5 and h is a function from E(G) to double-struck N so that h(e) ≥ 3 for all e ∈ E(G), or if Δ(G) ≥ 4 and h is a function from E(G) to double-struck N so that h(e) ≥ 4 for all e ∈ E(G).