Equitable Labelings of Corona Graphs

Abstract

A labeling $f$ of the vertices of a graph $G$ is said $k$-$equitable$ if each weight induced by $f$ on the edges of $G$ appears exactly $k$ times. A graph $G$ is said $equitable$ if for every proper divisor $k$ of its size, the graph $G$ has a $k$-equitable labeling.

A graph $G$ is a corona graph if $G$ is obtained from two graphs, $G_1$ and $G_2$, taking one copy of $G_1$, which is supposed to have order $p$, and $p$ copies of $G_2$, and then joining by an edge the $k^{th}$ vertex of $G_1$ to every vertex in the $k^{th}$ copy of $G_2$. We denote $G$ by $G_1\otimes G_2$.

In this paper, we proved that the corona graph $C_n\otimes K_1$ is equitable. Moreover, we show $k$-equitable labelings of the corona graph $C_m\otimes n{K}_1$, for some values of the parameters $k,m,$ and $n$.