On Antimagic Labelings of Disjoint Union of Complete $s$-Partite Graphs

Abstract

By an $(a,d)$-edge-antimagic total labeling of a graph $G(V,E)$, we mean a bijective function $f$ from $V(G)\cup E(G)$ onto the set $\{1,2,\dots ,|V(G)|+|E(G)|\}$ such that the set of all the edge-weights, $w(uv)=f(u)+f(uv)+f(v)$, for $uv\in E(G)$, is $\{a,a+d,a+2d,\dots ,a+(|E(G)|–1)d\}$, for two integers $a>0$ and $d\ge 0$.

In this paper, we study the edge-antimagic properties for the disjoint union of complete $s$-partite graphs.