Some classes of disconnected antimagic graphs and their joins

A labeling of a graph G is a bijection from E(G) to the set {1,2..., |E(G)|}. A labeling is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. We say that a graph is antimagic if it has an antimagic labeling. Hartsfield and Ringel conjectured in 1990 that every graph other than K ₂ is antimagic. In this paper, we show that the antimagic conjecture is false for the case of disconnected graphs. Furthermore, we find some classes of disconnected graphs that are antimagic and some classes of graphs whose complement are disconnected are antimagic.