Sparse graphs with vertex antimagic edge labelings

Hartsfeld and Ringel in 1990 introduced the concept of an antimagic labeling of a graph, that is, a vertex antimagic edge labeling and they also conjectured that every connected graph, except K2, is antimagic. As a means of providing an incremental advance towards proving the conjecture of Hartsfield and Ringel, in this paper we provide constructions whereby, given any degree sequence pertaining to a tree, we can construct two different vertex antimagic edge trees with the given degree sequence. Moreover, we modify a construction presented for trees to obtain an antimagic unicyclic graph with a given degree sequence pertaining to a unicyclic graph.