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

Let $G$ be a graph, $S$ be a set of vertices of $G$, and $λ(S)$ be the maximum number $\ell$ of pairwise edge-disjoint trees $T_1, T_2,..., T_{\ell}$ in $G$ such that $S\subseteq V(T_i)$ for every $1\leq i\leq \ell$. The generalized $k$-edge-connectivity $λ_k(G)$ of $G$ is defined as $λ_k(G)= min\{λ(S) | S\subseteq V(G) \ and \ |S|=k\}$. Thus $λ_2(G)=λ(G)$. In this paper, we consider the Nordhaus-Gaddum-type results for the parameter $λ_k(G)$. We determine sharp upper and lower bounds of $λ_k(G)+λ_k(\bar{G})$ and $λ_k(G)... λ_k(\bar{G})$ for a graph $G$ of order $n$, as well as for a graph of order $n$ and size $m$. Some graph classes attaining these bounds are also given.

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