Distance Two Labeling of Some Total Graphs

An L(2, 1)-labeling (or distance two labeling) of a graph 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(x, y) = 1 and |f(x) − f(y)| ≥ 1 if d(x, y) = 2. The L(2, 1)-labeling number λ(G) of G is the smallest number k such that G has an L(2, 1)-labeling with max{f(v) : v ∈ V (G)} = k. In this paper we completely determine the λ-number for total graphs of path Pn, cycle Cn, star K1,n and friendship graph Fn.