A sufficient condition for a graph to be weakly k-linked

Abstract For a pair ( s , t ) of vertices of a graph G , let λ G ( s , t ) denote the maximal number of edge-disjoint paths between s and t . Let ( s 1 , t 1 ), ( s 2 , t 2 ), ( s 3 , t 3 ) be pairs of vertices of G and k > 2. It is shown that if λ G ( s i , t i ) ≥ 2 k + 1 for each i = 1, 2, 3, then there exist 2 k + 1 edge-disjoint paths such that one joins s 1 and t 1 , another joins s 2 and t 2 and the others join s 3 and t 3 . As a corollary, every (2 k + 1)-edge-connected graph is weakly ( k + 2)-linked for k ≥ 2, where a graph is weakly k -linked if for any k vertex pairs ( s i , t i ), 1 ≤ i ≤ k , there exist k edge-disjoint paths P 1 , P 2 ,…, P k such that P i joins s i and t i for i = 1, 2,…, k .