On The Signed Edge Domination Number of Graphs

Details On The Signed Edge Domination Number of Graphs On The Signed Edge Domination Number of Graphs

2010-08-19

arxiv.org/abs/1008.3217v1

Discrete Mathematics. Vol. 309, Issue 3 (2009) pp. 587-594

Computer Science

Discrete Mathematics

Scientific paper

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 $n$.

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