Distance Magic Labelings of Complete Bipartite Graphs Obtained by Partition Modification

Graph labeling is an important area of graph theory that studies the assignment of integers to the vertices or edges of a graph according to specific rules. Among these labeling methods, distance magic labeling has attracted considerable attention due to its interesting combinatorial structure and applications in network design and communication systems. A distance magic labeling of a graph is a bijection from the vertex set to the set $\{1,2,\dots ,n\}$ such that the sum of the labels of the neighbors of each vertex is equal to a constant called the magic constant. This paper investigates the existence and construction of distance magic labelings for certain families of complete bipartite graphs. Two principal cases are studied, namely graphs of the form $K_{2m,2m}$ and $K_{2m-1,2m}$. For graphs of the form $K_{2m,2m}$, an explicit construction is developed showing that these graphs admit a distance magic labeling for every integer $m\ge 1$. The corresponding magic constant is derived as $k=m(4m+1)$ and the validity of the construction is verified by proving bijectivity of the labeling function and equality of vertex weights. A similar constructive approach is applied to graphs of the form $K_{2m-1,2m}$, where distance magic labelings are obtained using structured arithmetic label distributions. These constructions are further extended by applying vertex swapping techniques and block-based arguments to generate complete bipartite graphs $K_{p,q}$ that preserve the same magic constant for certain values of p and q. These findings contribute to the understanding of how arithmetic structure and partition properties influence the existence of distance magic labelings and suggest several directions for further research in graph labeling theory.