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

Abstract

Let $G$ be a graph, $S$ be a set of vertices of $G$, and $\lambda(S)$ 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 $\lambda_k(G)$ of $G$ is defined as $\lambda_k(G)= min\{\lambda(S) | S\subseteq V(G) \ and \ |S|=k\}$. 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 $\lambda_k(G)... \lambda_k(\bar{G})$ 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.