On Full Friendly Index Sets of Twisted Product of Mobius Ladders

Abstract

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A vertex labeling \(f: V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f^*: E(G) \to \mathbb{Z}_2\) defined by \(f^*(x,y) = f(x) + f(y)\), for each edge \((x,y) \in E(G)\). For each \(i \in \mathbb{Z}_2\), let \(v_f(i) = |\{v \in V(G) : f(v) = i\}|\) and \(e_f(i) = |\{e \in E(G) : f^*(e) = i\}|\). A vertex labeling \(f\) of a graph \(G\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined as \(\{|v_f(1) – v_f(0)| : \text{the vertex labeling } f \text{ is friendly}\}\). The full friendly index set of the graph \(G\), denoted by \(FFI(G)\), is defined as \(\{|e_f(1) – e_f(0)| : \text{the vertex labeling } f \text{ is friendly}\}\). In this paper, we determine \(FFI(G)\) and \(FI(G)\) for a class of cubic graphs which are twisted products of Möbius.