L(2,1)-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:uxuv1xuv2…xuvn−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). 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 dG(x,y)=1, and |f(x)−f(y)|≥1 if dG(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 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 N so that h(e)≥4 for all e∈E(G).