Nordhaus–Gaddum-type results for the generalized edge-connectivity of graphs

For a graph GG and a set SS of vertices of GG, let λ(S)λ(S) denote the maximum number ℓℓ of pairwise edge-disjoint Steiner trees T1,T2,⋯,TℓT1,T2,⋯,Tℓ in GG such that S⊆V(Ti)S⊆V(Ti) for every 1≤i≤ℓ1≤i≤ℓ. For an integer kk with 2≤k≤n2≤k≤n, where nn is the order of GG, the generalized kk-edge-connectivity λk(G)λk(G) of GG is defined as λk(G)=min{λ(S)∣S⊆V(G)and|S|=k}. In this paper, we consider the Nordhaus–Gaddum-type results for the parameter λk(G)λk(G). We obtain sharp upper and lower bounds of λk(G)+λk(G¯) and λk(G)⋅λk(G¯) for a graph GG of order nn, as well as a graph GG of order nn and size mm. Some graph classes attaining these bounds are also given.