On Friendly Index Sets of Generalized Books

Abstract

Let $G=(V,E)$ be a graph with a vertex labeling $f:V\to {\mathbb{Z}}_2$ that induces an edge labeling $f^{\ast }:E\to {\mathbb{Z}}_2$ defined by $f^{\ast }(xy)=f(x)+f(y)$. For each $i\in {\mathbb{Z}}_2$, let

$$v_f(i)=\text{card}\{v\in V:f(v)=i\}$$

and

$$e_f(i)=\text{card}\{e\in E:f^{\ast }(e)=i\}.$$

A labeling $f$ of a graph $G$ is said to be friendly if

$$|v_f(0)–v_f(1)|\le 1.$$

The friendly index set of $G$ is defined as

$$\{|e_f(1)–e_f(0)|:\text{the vertex labeling }f\text{ is friendly}\}.$$

In this paper, we determine the friendly index sets of generalized books.