Some Initial Results on the Supermagicness of Regular complete $k$-Partite Graphs

Abstract

Let G be a $(p,q)$-graph with p vertices and q edges. An edge-labeling assignment $\text{L : E}\to \text{N}$ is a map which assigns a positive integer to each edge in E. The induced map ${\text{L}}^{+}:\text{V}\to \text{N}$ defined by ${\text{L}}^{+}\text{(v)}=\mathrm{\Sigma }\{\text{L(u,v) : for all (u,v) in E}\}$ is called the vertex sum. The edge labeling assignment is called \underline{magic} if ${\text{L}}^{+}$ is a constant map. If L is a bijection with $\text{L(E)}=\{1,2,\dots ,\text{q}\}$ and L is magic then we say L is supermagic. B. M. Stewart showed that ${\text{K}}_5$ is not supermagic and when $\text{n}\equiv 0(\mathrm{mod}4)$ , ${\text{K}}_{\text{n}}$ is not supermagic. In this paper, we exhibit supermagicness for a class of regular complete k-partite graphs.