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

Abstract

Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). An edge labeling \(f: E \to Z_{2}\) induces a vertex labeling \(f^{+} : V \to Z_{2}\) defined by \( f^{+}(v) \equiv \sum_{uv \in E} f(uv) \pmod 2 \), for each vertex \(v \in V\). For \(i \in Z_{2}\), let \( v_{f}(i) = |\{v \in V : f^+(v) = i\}| \) and \( e_{f}(i) = |\{e \in E : f(e) = i\}| \). An edge labeling \(f\) of a graph \(G\) is said to be edge-friendly if \( |e_{f}(1) – e_{f}(0)| \le 1 \). The set \(\{v_f(1) – v_f(0) : f \text{ is an edge-friendly labeling of } G\}\) 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.