Caterpillars are Antimagic

Abstract

An antimagic labeling of a graph G is a bijection from the set of edges E(G) to $$\{1,2,\ldots ,|E(G)|\}$$ , such that all vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of the labels assigned to the edges incident to u. A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than $$K_2$$ is antimagic and the conjecture remains open even for trees. Here, we prove that caterpillars are antimagic by means of an $$O(n \log n)$$ algorithm.