Strength and fractional arboricity of complementary graphs Original Research Article

Let G be a simple graph and Gc be the complement of G. Let ω(G) denote the number of components of G. As in Catlin et al. (1992), for a nontrivial graph G, the strength of G is where the minimum is over all subsets S ⊆ E(G) such that ω(G − S) > ω(G). The fractional arboricity of a nontrivial graph G is where the maximum runs over all subgraphs H with | V(H) | > ω(H). In this note, we shall present Nordhaus-Gaddum types of inequalities on the strength and the fractional arboricity of a graph G. In particular, we show that if G is a simple graph on n ⩾ 4 vertices, then each of the following holds: where d ϵ [γo], [γo] + 1 in (e) and (f), and. Moreover, all the bounds are best possible.