Skolem-Gracefulness of \(k\)-Stars

Abstract

A graph \(G = (V, E)\) is Skolem-graceful if its vertices can be labelled \(1, 2, \ldots, |V|\), so that the edges are labelled \(1, 2, \ldots, |E|\), where each edge label is the absolute difference of the labels of the two end-vertices. It is shown that a \(k\)-star is Skolem-graceful only if at least one star has even size or \(k \equiv 0\) or \(1 \pmod{4}\), and for \(k \leq 5\), a \(k\)-star is Skolem-graceful if at least one star has even size or \(k \equiv 0\) or \(1 \pmod{4}\). In this paper, we show that \(k\)-stars are Skolem-graceful if at least one star has even size or \(k \equiv 0\) or \(1 \pmod{4}\) for all positive integer \(k\).