Bimagic Vertex Labelings

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 K n , n , m . It is proved that the sequence of bi-regular graphs K n ( ij ) = (( K n − 1 − M ) + K 1 ) − ( u n u i ) − ( u n u j ) admits 1-vertex bimagic vertex labeling, where u i , u j is any pair of non-adjacent vertices in the graph K n − 1 − M , u n is a vertex of K 1 , M is perfect matching of the complete graph K n − 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 + n 2 and ( n + 1)( n + r )/2 + nr . Two new types of graphs that do not admit 1-vertex bimagic vertex labelings are defined.