The notion of the equivalence of vertex labelings on a given graph is introduced. The equivalence of three bimagic labelings for regular graphs is proved. A particular solution is obtained for the problem of the existence of a 1-vertex bimagic vertex labeling of multipartite graphs, namely, for graphs isomorphic with Kn, n, m. It is proved that the sequence of bi-regular graphs Kn(ij) = ((Kn − 1 − M) + K1) − (unui) − (unuj) admits 1-vertex bimagic vertex labeling, where ui, uj is any pair of non-adjacent vertices in the graph Kn − 1 − M, un is a vertex of K1, M is perfect matching of the complete graph Kn − 1. It is established that if an r-regular graph G of order n is distance magic, then graph G + G has a 1-vertex bimagic vertex labeling with magic constants (n + 1)(n + r)/2 + n2 and (n + 1)(n + r)/2 + nr. Two new types of graphs that do not admit 1-vertex bimagic vertex labelings are defined.