On the signed edge domination number of graphs

Let γ 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 ≥ 2 ) , γ s ′ ( G ) ≥ 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 γ s ′ ( G ) ≤ − m 6 | V ( G ) | . Also for every two natural numbers m and n , we determine γ s ′ ( K m , n ) , where K m , n is the complete bipartite graph with part sizes m and n .