On edge-disjoint spanning trees with small depths

Abstract

Let $\text{G}$ be a graph with $\text{n}$ vertices. Suppose that there are $\text{k}$ edge-disjoint spanning trees in $\text{G}$ . We show that if the minimum degree of the vertices in $\text{G}$ is at least $\text{kl}$ , then there are $\text{k}$ edge-disjoint spanning trees rooted at any vertex with depth $\text{O}\text{(n/l)}$ in $\text{G}$ .

Also, we show that given a graph, a vertex $\text{r}$ of the graph, and a positive integer $\text{k}$ , the problem of finding $\text{k}$ edge-disjoint spanning trees rooted at $\text{r}$ with optimal depth is NP-hard.