On the $E_3$-cordiality of some Graphs

Abstract

The definition of $E_k$-cordial graphs is advanced by Cahit and Yilmaz${}^{[1]}$. Based on [1], a graph $G$ is said to be $E_3$-cordial if it is possible to label the edges with the numbers from the set $\{0,1,2\}$ in such a way that, at each vertex $v$, the sum of the labels on the edges incident with $v$ modulo $3$ satisfies the inequalities $|v(i)–v(j)|\le 1$ and $|e(i)–e(j)|\le 1$, where $v(s)$ and $e(t)$ are, respectively, the number of vertices labeled with $s$ and the number of edges labeled with $t$. In [1]-[3], authors discussed the $E_3$-cordiality of $P_n$ $(n\ge 3)$; stars $S_n$, $|S_n|=n+1$; $K_n$ $(n\ge 3)$, $C_n$ $(n\ge 3)$, the one point union of any number of copies of $K_n$ and $K_m\odot K_m$. In this paper, we give the $E_3$-cordiality of $W_n$, $P_m\times P_n$, $K_{m,n}$ and trees.