Links in edge-colored graphs

A graph is k -linked ( k -edge-linked), k ≥ 1 , if for each k pairs of vertices x 1 , y 1 , … , x k , y k , there exist k pairwise vertex-disjoint (respectively edge-disjoint) paths, one per pair x i and y i , i = 1 , 2 , … , k . Here we deal with the properly edge-colored version of the k -linked ( k -edge-linked) problem in edge-colored graphs. In particular, we give conditions on colored degrees and/or number of edges, sufficient for an edge-colored multigraph to be k -linked ( k -edge-linked). Some of the results obtained are the best possible. Related conjectures are proposed.