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 $\{0,\dots ,k-1\}$. 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 $\{0,\dots ,k-1\}$ equitably to the vertices as well as edges.

If $G_1,\dots ,G_T$ 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 $G_1,\dots ,G_T$.

In this paper we prove that the ${\overline{K}}_n$-union of gears is edge-$3$-equitable.