On Friendly Index Sets of Root-unions of Stars By Cycles

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A labeling $f:V(G)\to {\mathbb{Z}}_2$ induces an edge labeling $f^{\ast }:E(G)\to {\mathbb{Z}}_2$, defined by $f^{\ast }(xy)=f(x)+f(y)$, for each edge $xy\in E(G)$. For $i\in {\mathbb{Z}}_2$, let ${\text{v}}_f(i)=\text{card}\{v\in V(G):f(v)=i\}$ and ${\text{e}}_f(i)=\text{card}\{e\in E(G):f^{\ast }(e)=i\}.$A labeling $f$ of a graph $G$ is said to be friendly if $|{\text{v}}_f(0)–{\text{v}}_f(1)|\le 1.$The friendly index set of the graph $G$, $FI(G)$, is defined as $\{|{\text{e}}_f(0)–{\text{e}}_f(1)|:\text{the vertex labeling }f\text{ is friendly}\}.$This is a generalization of graph cordiality. We introduce a graph construction called the root-union and investigate when gaps exist in the friendly index sets of root-unions of stars by cycles.