Edge-three Cordial Graphs Arising from Complete Graphs

A graph \( G \) is said to be \( E_k \)-Cordial if there is an edge labeling \( f : E(G) \rightarrow \{0,1,\ldots,k-1\} \) such that, at each vertex \( v \), the sum modulo \( k \) of the labels on the edges incident with \( v \) is \( f(v) \) and it satisfies the inequalities \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \), where \( v_f(s) \) and \( e_f(t) \) are, respectively, the number of vertices labeled with \( s \) and the number of edges labeled with \( t \). The map \( f \) is then called an \( E_k \)-cordial labeling of \( G \).

This paper investigates \( E_3 \)-cordiality of snakes, one point unions, path unions, and coronas involving complete graphs.