On Antimagic Labelings of Disjoint Union of Complete \(s\)-Partite Graphs

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 \( d \geq 0 \).

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