Links in edge-colored graphs

A graph is kk-linked (kk-edge-linked), k≥1k≥1, if for each kk pairs of vertices x1,y1,…,xk,ykx1,y1,…,xk,yk, there exist kk pairwise vertex-disjoint (respectively edge-disjoint) paths, one per pair xixi and yiyi, i=1,2,…,ki=1,2,…,k. Here we deal with the properly edge-colored version of the kk-linked (kk-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 kk-linked (kk-edge-linked). Some of the results obtained are the best possible. Related conjectures are proposed.