On the Edge-Balance Index Sets of $n$-Wheels

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A labeling $f$ of a graph $G$ is said to be edge-friendly if $|e_f(0)–e_f(1)|\le 1$, where $e_f(i)=\text{card}\{e\in E(G):f(e)=i\}$. An edge-friendly labeling $f:E(G)\to {\mathbb{Z}}_2$ induces a partial vertex labeling $f^{+}:V(G)\to A$ defined by $f^{+}(x)=0$ if the edges incident to $x$ are labeled $0$ more than $1$. Similarly, $f^{+}(x)=1$ if the edges incident to $x$ are labeled $1$ more than $0$. $f^{+}(x)$ is not defined if the edges incident to $x$ are labeled $1$ and $0$ equally. The edge-balance index set of the graph $G$, $EBI(G)$, is defined as $\{|v_f(0)–v_f(1)|:\text{the edge labeling }f\text{ is edge-friendly}\}$, where $v_f(i)=\text{card}\{v\in V(G):f^{+}(v)=i\}$.

An $n$-wheel is a graph consisting of $n$ cycles, with each vertex of the cycles connected to one central hub vertex. The edge-balance index sets of $n$-wheels are presented.