Paths and edge-connectivity in graphs III. Six-terminalk paths

Suppose thatk ≥ 1 is an odd integer, (s ₁,t ₁),..., (s _k> ,t _k ) are pairs of vertices of a graphG andλ(s _i ,t _i ) is the maximal number of edge-disjoint paths betweens _i andt _i . We prove that ifλ(s _i ,t _i )≥ k (1≤ i ≤ k) and |{s ₁,...s _k ,t ₁,...,t _k }| ≤ 6, then there exist edge-disjoint pathsP ₁,...,P _k such thatP _i has endss _i andt _i (1≤ i ≤ k).