Multiple weak 2-linkage and its applications on integer flows of signed graphs

For two pairs of vertices x1,y1 and x2,y2, Seymour and Thomassen independently presented a characterization of graphs containing no edge-disjoint (x1,y1)-path and (x2,y2)-path. In this paper we first generalize their result to k≥2 pairs of vertices. Namely, for 2k vertices x1,y1,x2,y2,…,xk,yk, we characterize the graphs without edge-disjoint (xi,yi)-path and (xj,yj)-path for any 1≤i