Some bounds for the signed edge domination number of a graph

The closed neighbourhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end vertex with e. Let f be a function on the edges of G into the set {−1, 1}. If e∈NG[x] f(e) ≥ 1 for every x ∈ E(G), then f is called a signed edge domination function of G. The minimum value of ∑ x∈E(G) f(x), taken over every signed edge domination function f of G, is called signed edge domination number of G and denoted by γ′ s(G). It has been proved that γ ′ s(G) ≥ n − m ∗ Corresponding author. S. AKBARI ET AL. /AUSTRALAS. J. COMBIN. 58 (1) (2014), 60–66 61 for every graph G of order n and size m. In this paper we prove that γ′ s(G) ≥ 2α ′(G)−m 3 for every simple graph G, where α′(G) is the size of a maximum matching of G. We also prove that for a simple graph G of order n whose each vertex has an odd degree, γ′ s(G) ≤ n− 2α ′(G) 3 .