Full Edge-Friendly Index Sets of One Point Union of Cycles

Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). An edge labeling \(f: E → Z₂\) induces a vertex labeling \(f⁺ : V → Z₂\) defined by \( f⁺(v) ≡ ∑uv ∈ E f(uv) 2 \), for each vertex \(v ∈ V\). For \(i ∈ Z₂\), let \( vf(i) = |\{v ∈ V : f^+(v) = i\}| \) and \( ef(i) = |\{e ∈ E : f(e) = i\}| \). An edge labeling \(f\) of a graph \(G\) is said to be edge-friendly if \( |ef(1) – ef(0)| ≤ 1 \). The set \(\{v_f(1) – v_f(0) : f is an edge-friendly labeling of G\}\) is called the full edge-friendly index set of \(G\). In this paper, we shall determine the full edge-friendly index sets of one point union of cycles.