Labeling trees of small diameters with consecutive integers (Q6110572)

Given a simple graph \(G=(V,E)\) and an integer \(k\), the authors are looking for a bijection \(f\) from \(E\) to \(\{k+1, \ldots, k+m\}\) such that for each vertex \(v\) the sum of \(f(e)\) over all edges \(e\) incident to \(v\) is unique. If such a bijection exists, they call \(G\) \(k\)-shifted antimagic, hereby generalizing the notion of an antimagic graph proposed by \textit{N. Hartsfield} and \textit{G. Ringel} [Pearls in graph theory. A comprehensive introduction. Rev. and augmented ed. Orlando, FL: Academic Press (1994; Zbl 0823.05001)]. In this paper, it is shown that every tree of diameter four or five is \(k\)-shifted antimagic for every integer \(k\), with the exception of two previously known examples.