On Edge-\(3\)-Equitability of \(K_{n}\)-Union of Gears.

Abstract

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 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 we prove that the \( \overline{K}_n \)-union of gears is edge-\( 3 \)-equitable.