On the \(E_{3}\)-cordiality of some Graphs

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 \( |v(i) – v(j)| \leq 1 \) and \( |e(i) – e(j)| \leq 1 \), 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 \( P_n \) \( (n \geq 3) \); stars \( S_n \), \( |S_n| = n + 1 \); \( K_n \) \( (n \geq 3) \), \( C_n \) \( (n \geq 3) \), 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.