The edge balance properties of cubic graphs

For a simple, undirected graph $G(V,E)$, let $f$ be an edge labeling $f$: $E\to \{0,1\}$ such that, for any vertex $v$ and $i\in \{0,1\}$, in the edges incident on $v$, if the number of the edges labeled $i$ is more than the number of edges labeled $1-i$, then the label of $v$ is defined by $i$; $v$ is not defined otherwise. In a labeling graph of $G$, let $i\in \{0,1\}$, $e_f(i)=|\{e\in E:f(e)=i\}|$ and $v_f(i)=|\{v\in V:$ the label of $v$ is $i\}|$. After $f$ runs over all edge labelings satisfying $|e_f(1)-e_f(0)|\le 1$, the set $\{|v_f(1)-v_f(0)|:|e_f(1)-e_f(0)|\le 1\}$ is called the edge-balance index set of graph $G$, denoted by $EBI(G)$. In this paper, the set $\{v_f(1)-v_f(0):|e_f(1)-e_f(0)|\le 1\}$ is called the full edge-balance index set of graph $G$, denoted by $FEBI(G)$. Some results are obtained on $FEBI(G)$, and the relationship between $FEBI(G)$ and $EBI(G)$ is discussed. By finding some closed trails, the $FEBI$ and $EBI$ of some classes of cubic graphs are obtained.