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 f ℓ( k)( g ℓ( k)) such that for any f ℓ( k)-edge-connected ( g ℓ( k)-edge-connected) vertex set A of a graph G with | A|⩽ℓ(| V( G)− A|⩽ℓ) there exists a system T of k edge-disjoint trees such that A⊆ V( T) for each T∈ T . We determine f 3(k)=⌊ 8k+3 6 ⌋ . Furthermore, we determine for all natural numbers ℓ, k the smallest number f ℓ ∗(k) such that every f ℓ ∗(k) -edge-connected graph on at most ℓ vertices contains a system of k edge-disjoint spanning trees, and give applications to line graphs.