Edge Friendly Labelings of Graph -\( I\)

Abstract

Let \(G = (V, E)\) be a graph. An edge labeling \(f: E \to \mathbb{Z}_2\) induces a vertex labeling \(f^*: V \to \mathbb{Z}_2\) defined by \(f^*(v) = \sum_{uv \in E} f(uv) \pmod{2}\). For each \(i \in \mathbb{Z}_2\), define \(E_i(f) = |f^{-1}(i)|\) and \(V_i(f) = |(f^*)^{-1}(i)|\). We call \(f\) edge-friendly if \(|E_1(f) – E_0(f)| \leq 1\). The edge-friendly index \(I_f(G)\) is defined as \(V_1(f) – V_0(f)\), and the full edge-friendly index set \(FEFI(G)\) is defined as \(\{I_f(G): f \text{ is an edge-friendly labeling}\}\). Further, the edge-friendly index set \(EFI(G)\) is defined as \(\{|I_f(G)|: f \text{ is an edge-friendly labeling}\}\). In this paper, we study the full edge-friendly index set of the star \(K_{1,n}\), \(2\)-regular graph, wheel \(W_n\), and \(m\) copies of path \(mP_n\), \(m \geq 1\).