L(p,q)-labelings of subdivisions of graphs

Given a graph G and a function h from E(G) to 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_uvⁿ⁻¹v, where n=h(uv). When h(e)=c is a constant for all e∈E(G), we use G_(c) to replace G_(h). For a given graph G, an L(p,q)-labeling of G is a function f from the vertex set V(G) to the set of all nonnegative integers such that f(u)−f(v)≥p if d_G(u,v)=1, and f(u)−f(v)≥q if d_G(u,v)=2. A k-L(p,q)-labeling is an L(p,q)-labeling such that no label is greater than k. The L(p,q)-labeling number of G, denoted by λ_p,q(G), is the smallest number k such that G has a k-L(p,q) -labeling. We study the L(p,q)-labeling numbers of subdivisions of graphs in this paper. We prove that λ_p,q(G₍₃₎)=p+(Δ−1)q when p≥2q and [Formula presented], and show that λ_p,q(G_(h))=p+(Δ−1)q when p≥2q and [Formula presented], where h is a function from E(G) to N so that h(e)≥3 for all e∈E(G).