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 $\lambda (S)$ be the maximum number $\mathrm{\ell }$ of pairwise edge-disjoint trees $T_1,T_2,\mathrm{.}\mathrm{.}\mathrm{.},T_{\mathrm{\ell }}$ in $G$ such that $S\subseteq V(T_i)$ for every $1\le i\le \mathrm{\ell }$. The generalized $k$-edge-connectivity ${\lambda }_k(G)$ of $G$ is defined as ${\lambda }_k(G)=min\{\lambda (S)\mathrm{\mid }S\subseteq V(G)and\mathrm{\mid }S\mathrm{\mid }=k\}$. Thus ${\lambda }_2(G)=\lambda (G)$. In this paper, we consider the Nordhaus-Gaddum-type results for the parameter ${\lambda }_k(G)$. We determine sharp upper and lower bounds of ${\lambda }_k(G)+{\lambda }_k(\bar{G})$ and $\lambda_k(G)… \lambda_k(\bar{G})$foragraph$G$oforder$n$, as well as for a graph of order $n$ and size $m$. Some graph classes attaining these bounds are also given.