Upper bounds on the signed edge domination number of a graph

A signed edge domination function (or SEDF) of a simple graph $G=(V,E)$ is $f: E\rightarrow \{1,-1\}$ such that $\sum_{e'\in N[e]}f(e')\ge 1$ holds for each $e\in E$, where $N[e]$ the set edges in $G$ share at least one endpoint with $e$. Let $\gamma_s'(G)$ denote minimum value $f(G)$ among all SEDFs $f$, $f(G)=\sum_{e\in E}f(e)$.In 2005, Xu conjectured $\gamma_s'(G)\le n-1$, $n$ order $G$. This conjecture has been proved two cases $v_{odd}(G)=0$ and $v_{even}(G)=0$, $v_{odd}(G)$ (resp. $v_{even}(G)$) number odd even) vertices article proves Xu's $v_{even}(G)\in \{1, 2\}$. We also show any $n$, n+v_{odd}(G)/2$ n-2+v_{even}(G)$ when $v_{even}(G)>0$, thus (4n-2)/3$. Our result improves best current upper bound \lceil 3n/2\rceil$.