Caterpillars with maximum degree 3 are antimagic

Abstract

Let G=(V,E) be a connected graph with m edges. An antimagic labeling of G is a one-to-one mapping from E to {1,2,…,m} such that the vertex sum (i.e., sum of the labels assigned to edges incident to a vertex) for distinct vertices are different. A graph G is called antimagic if G has an antimagic labeling. It was conjectured by Hartsfield and Ringel that every tree other than K2 is antimagic. The conjecture remains open though it was verified for trees with some constrains. Caterpillars are an important subclass of trees. This paper shows caterpillars with maximum degree 3 are antimagic, which gives an affirmative answer to an open problem of Lozano et al. (2019).