On The Signed Edge Domination Number of Graphs

TL;DR

This paper investigates the signed edge domination number in graphs, disproving a 2006 conjecture and providing bounds for specific graph classes, including complete bipartite graphs.

Contribution

It disproves Xu's conjecture that all 2-connected graphs have a signed edge domination number at least 1 and determines this number for complete bipartite graphs.

Findings

Disproved Xu's conjecture for 2-connected graphs.

Established bounds for the signed edge domination number in m-connected graphs.

Calculated the exact signed edge domination number for complete bipartite graphs.

Abstract

Let $\gamma'_s(G)$ be the signed edge domination number of G. In 2006, Xu conjectured that: for any $2$-connected graph G of order $ n (n \geq 2),$ $\gamma'_s(G)\geq 1$. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer $m$, there exists an $m$-connected graph $G$ such that $ \gamma'_s(G)\leq -\frac{m}{6}|V(G)|.$ Also for every two natural numbers $m$ and $n$, we determine $\gamma'_s(K_{m,n})$, where $K_{m,n}$ is the complete bipartite graph with part sizes $m$ and $n$.