Antipodal Labelings for Cycles

Abstract

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 \(v\), \(|f(u) — f(v)| \geq d — d(u,v)\), where \(d(u, v)\)
is the distance between \(u\) and \(v\). The span of an antipodal labeling f is \(\max{f(u) — f(v) : u,v \in V(G)}\). The
antipodal number for G, denoted by an\((G)\), is the minimum span of an antipodal labeling for \(G\). Let \(C_n\) denote
the cycle on n vertices. Chartrand \(et al\). \([4]\) determined the value of an\((C_n)\) for \(n \equiv 2 \pmod 4\). In this article we
obtain the value of an\((C_n)\) for \(n \equiv 1 \pmod 4\), confirming a conjecture in \([4]\). Moreover, we settle the case \(n \equiv 3 \pmod 4\), and improve the known lower bound and give an upper bound for the case \(n \equiv 0 \pmod 4\).