PACKING EDGE-DISJOINT ODD EULERIAN SUBGRAPHS THROUGH PRESCRIBED VERTICES IN 4-EDGE-CONNECTED GRAPHS

In this paper, we show the Erdos-Posa property for edge-disjoint packing of S-closed walks with parity constraints in 4-edge-connected graphs. More precisely, we prove that for any 4-edge-connected graph G and any vertex subset S, either G has k edge-disjoint elementary closed odd walks, each of which has at least one vertex of S, or G has an edge set F with vertical bar F vertical bar <= f(k) such that G-F has no such walks. The 4-edge-connectivity is the best possible in the sense that 3-edge-connected graphs do not satisfy the statement. Since the proof is constructive, we can design a fixed-parameter algorithm for finding k edge-disjoint walks satisfying the conditions in a 4-edge connected graph for a parameter k. In addition, this gives a simple fixed-parameter algorithm for the parity edge-disjoint walks problem with k terminal pairs.