\(E_3\)-Cordiality of Some Helm-Related Graphs

Abstract

A \(k\)-edge labeling of a graph \(G\) is a function \(f: E(G) \to \{0, \ldots, k-1\}\). Such a labeling induces a labeling on the vertex set \(V(G)\) by defining \(f(v) := \sum f(e) \pmod{k}\), where the summation is taken over all edges \(e\) incident on \(v\). For an edge labeling \(f\), let \(v_f(i)\) (resp., \(e_f(i)\)) denote the number of vertices (resp., edges) receiving the label \(i\). A graph \(G\) is said to be \(E_k\)-cordial if there exists a \(k\)-edge labeling \(f\) of \(G\)such that \(|v_f(i) – v_f(j)| \leq 1\) and \(|e_f(i) – e_f(j)| \leq 1\) for all \(0 \leq i, j \leq k-1\). A wheel \(W_n\) is the join of the cycle \(C_n\) on \(n\) vertices and \(K_1\). A Helm \(H_n\) is obtained by attaching a pendent edge to each vertex of the cycle of the wheel \(W_n\). We prove that (i) Helms, (ii) one-point unions of helms, and (iii) path unions of helms are \(E_3\)-cordial.