Edge-disjoint trees containing some given vertices in a graph

Abstract

We show that for any two natural numbers k,ℓ there exist (smallest natural numbers fℓ(k)(gℓ(k)) such that for any fℓ(k)-edge-connected (gℓ(k)-edge-connected) vertex set A of a graph G with A ≤ℓ( V(G) - A ≤ℓ) there exists a system of k edge-disjoint trees such that A⊆V(T) for each T ∈. We determine f3(k) = ⌊8k+3/6⌋. Furthermore, we determine for all natural numbers ℓ,k the smallest number, fℓ*(k) such that every fℓ*(k -edge-connected graph on at most ℓ vertices contains a system of k edge-disjoint spanning trees, and give applications to line graphs. © 2003 Published by Elsevier Science (USA).