Abstract. 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 of isomorphic K_n,n,m. It is proved that the sequence of bi-regular graphs K_{n ( i, j )} = ((K_n –1– M )+K₁)–(u_nu_i )–(u_nu_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 the vertex of K₁, M is the perfect matching of the complete graph K_n – 1. It is established that if the r-regular graph G of order n is distance magic one, then the 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.

Keywords: distance magic labeling, 1-vertex bimagic vertex labeling, odd 1-vertex bimagic vertex labeling, even 1-vertex bimagic vertex labeling.

¹ Flight Academy of the National Aviation University, Kropyvnytskyi, Ukraine

 marina_semenyuta@ukr.net

² Flight Academy of the National Aviation University, Kropyvnytskyi, Ukraine

 nvn60@ukr.net

³ Flight Academy of the National Aviation University, Kropyvnytskyi, Ukraine

 nvn60@ukr.net