Antimagic labeling of some biregular bipartite graphs

An antimagic labeling of a graph G = (V,E) is a one-to-one mapping from E to {1, 2, . . . , |E|} such that distinct vertices receive different label sums from the edges incident to them. G is called antimagic if it admits an antimagic labeling. It was conjectured that every connected graph other than K2 is antimagic. The conjecture remains open though it was verified for several classes of graphs such as regular graphs. A bipartite graph is called (k, k′)-biregular, if each vertex of one of its parts has the degree k, while each vertex of the other parts has the degree k′. This paper shows the following results. (1) Each connected (2, k)-biregular (k ≥ 3) bipartite graph is antimagic; (2) Each (k, pk)-biregular (k ≥ 3, p ≥ 2) bipartite graph is antimagic; (3) Each (k, k + y)-biregular (k ≥ 3, y ≥ 1) bipartite graph is antimagic.