On antimagic directed graphs

An antimagic labeling of an undirected graph G with n vertices and m edges is a bijection from the set of edges of G to the integers {1,...,m} such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labeling. In (N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston, 1990, pp. 108-109), Hartsfield and Ringel conjectured that every simple connected graph, other than K ₂, is antimagic. Despite considerable effort in recent years, this conjecture is still open. In this article we study a natural variation; namely, we consider antimagic labelings of directed graphs. In particular, we prove that every directed graph whose underlying undirected graph is "dense" is antimagic, and that almost every undirected d-regular graph admits an orientation which is antimagic.