Antipodal Labelings for Cycles

Let G be a graph with diameter d. An antipodal labeling of G is a function f that assigns to each vertex a non-negative integer (label) such that for any two vertices u and v, it is satisfied that |f(u) − f(v)| ≥ d − d(u, v), where d(u, v) is the distance between u and v. The span of an antipodal labeling f is max{f(u) − f(v) : u, v ∈ V (G)}. The antipodal number for G, denoted by an(G), is the minimum span of an antipodal labeling for G. Let Cn denote the cycle on n vertices. Chartrand, Erwin and Zhang [4] determined the value of an(Cn) for n ≡ 2 (mod 4). In this article we determine the value of an(Cn) for n ≡ 1 (mod 4), confirming a conjecture of Chartrand el al. Moreover, we settle the case n ≡ 3 (mod 4); and improve the known lower bound and give an upper bound for the case n ≡ 0 (mod 4).