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A \( k \)-edge labeling of a graph \( G \) is a function \( f \) from the edge set \( E(G) \) to the set of integers \(\{0, \ldots, k-1\}\). Such a labeling induces a labeling \( f \) on the vertex set \( V(G) \) by defining \( f(v) = \sum f(e) \), where the summation is taken over all the edges incident on the vertex \( v \) and the value is reduced modulo \( k \). Cahit calls this labeling edge-\( k \)-equitable if \( f \) assigns the labels \(\{0, \ldots, k-1\}\) equitably to the vertices as well as edges.
If \( G_1, \ldots, G_T \) is a family of graphs each having a graph \( H \) as an induced subgraph, then by \( H \)-union \( G \) of this family we mean the graph obtained by identifying all the corresponding vertices as well as edges of the copies of \(H\) in \(G_1, \ldots, G_T\).
In this paper, which is a sequel to the paper entitled `On edge-\(3\)-equitability of \(\overline{K}_n\)-union of gears’, we prove that \(\overline{K}_n\)-union of copies of helm \(H_n\) is edge-\(3\)-equitable for all \(n \geq 6\).