Signed edge domination numbers of complete tripartite graphs: Part 2

Abstract

The closed neighborhood $N_G[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having an end-vertex in common with $e$. Let $f$ be a function on $E(G)$, the edge set of $G$, into the set $\{-1,1\}$. If ${\sum }_{x\in N[e]}f(x)\ge 1$ for each edge $e\in E(G)$, then $f$ is called a signed edge dominating function of $G$. The signed edge domination number of $G$ is the minimum weight of a signed edge dominating function of $G$. In this paper, we find the signed edge domination number of the complete tripartite graph $K_{m,n,p}$, where $1\le m\le n$ and $p\ge m+n$. This completes the search for the signed edge domination numbers of the complete tripartite graphs.

arXiv comment21 pages