Balanced degree-magic complements of bipartite graphs

A graph G is called degree-magic if it admits a labelling of the edges by integers 1,2,…,|E(G)| such that the sum of the labels of the edges incident with any vertex v is equal to 1+|E(G)|2deg(v). Degree-magic graphs extend supermagic regular graphs. In this paper we present a construction of some degree-magic complements of bipartite graphs.