Bounds of the signed edge domination number of complete multipartite graphs

Abstract:

      A function $f: E(G)\rightarrow\{-1,1\}$ is called a \emph{signed edge dominating function} (SEDF for short) of $G$ if $f[e]=f(N[e])= \sum_{e'\in N[e]}f(e')\geq1$, for every edge $e\in E(G)$. The \emph{signed edge domination number} $\gamma_{s}\,'(G)$ of $G$ is the minimum weight among all signed edge dominating functions of $G$. In this paper, we initiate the study of this parameter for $G$ a complete multipartite graph. we provide the lower and upper bounds of $\gamma_{s}\,'(G)$ for $G$ a complete $r$-partite graph with $r$ even and all parts equal.