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Suppose \( G \) is a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A \) be an additive Abelian group. A vertex labeling \( f: V(G) \to A \) induces an edge labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) + f(y) \). For \( a \in A \), let \( n_a(f) \) and \( m_a(f) \) be the number of vertices \( v \) and edges \( e \) with \( f(v) = a \) and \( f^*(e) = a \), respectively. A graph \( G \) is \( A \)-cordial if there exists a vertex labeling \( f \) such that \( |n_a(f) – n_b(f)| \leq 1 \) and \( |m_a(f) – m_b(f)| \leq 1 \) for all \( a, b \in A \). When \( A = \mathbb{Z}_k \), we say that \( G \) is \( k \)-cordial instead of \( \mathbb{Z}_k \)-cordial. In this paper, we investigate certain regular graphs and ladder graphs that are \( 4 \)-cordial and we give a complete characterization of the \( 4 \)-cordiality of the complete \( 4 \)-partite graph. An open problem about which complete multipartite graphs are not \( 4 \)-cordial is given.