Spiders are Antimagic

Abstract

For a graph \(G\), an edge labeling of \(G\) is a bijection \(f: E(G) \to \{1, 2, \ldots, |E(G)|\}\). The \emph{induced vertex sum} \(f^*\) of \(f\) is a function defined on \(V(G)\) given by \(f^+(u) = \sum_{uv \in E(G)} f(uv)\) for all \(u \in V(G)\). A graph \(G\) is called \emph{antimagic} if there exists an edge labeling of \(G\) such that the induced vertex sum of the edge labeling is injective. Hartsfield and Ringel conjectured in 1990 that all connected graphs except \(K_2\) are antimagic. A spider is a connected graph with exactly one vertex of degree exceeding \(2\). This paper shows that all spiders are antimagic.