On Friendly Index Sets of Cycles with Parallel Chords

Abstract

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A\) be an abelian group. A labeling \(f : V(G) \to A\) induces an edge labeling \(f^* : E(G) \to A\) defined by \(f^*(xy) = f(x) + f(y)\), for each edge \(xy \in E(G)\). For \(i \in A\), let \(v_f(i) = \mathrm{card}\{v \in V(G) : f(v) = i\}\) and \(e_f(i) = \mathrm{card}\{e \in E(G) : f^*(e) = i\}\). Let \(c(f) = \{|e_f(i) – e_f(j)|: (i, j) \in A \times A\}\). A labeling \(f\) of a graph \(G\) is said to be \(A\)-friendly if \(|v_f(i)- v_f(j)| \leq 1\) for all \((i, j) \in A \times A\). If \(c(f)\) is a \((0, 1)\)-matrix for an \(A\)-friendly labeling \(f\), then \(f\) is said to be \(A\)-cordial. When \(A = \mathbb{Z}_2\), the friendly index set of the graph \(G\), \(FI(G)\), is defined as \(\{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\}\). In [13] the friendly index set of cycles are completely determined. In this paper we describe the friendly index sets of cycles with parallel chords. We show that for a cycle with an arbitrary non-empty set of parallel chords, the numbers in its friendly index set form an arithmetic progression with common difference 2.