Packing odd T‐joins with at most two terminals

Abstract

We prove the Cycling conjecture for the class of clutters of odd T -joins with at most two terminals. Corollaries of this result include, the characterization of weakly bipartite graphs [5], packing two-commodity paths [7,10], packing T -joins for |T | ≤ 4, and a new result on covering edges with cuts.

Clean Clutters and Dyadic Fractional Packings

“…Given a signed graph, if the (binary) clutter of odd circuits is ideal, then the clutter must have an optimal fractional packing that is 1 2 -integral [21]. More generally, given a signed graph and (possibly equal) vertices s, t, if the (binary) clutter of odd st-walks is ideal, then the clutter must have an optimal fractional packing that is 1 2 -integral [9,10]. We have considered ideal clutters without an intersecting minor, as well as ideal binary clutters.…”

A survey on flows in graphs and matroids

“…Similarly, the Max Work Min Potential Theorem of Duffin [8] implies the Cycling conjecture for e-paths of co-graphic matroids. We recently proved the following [1,2], Theorem 22. The Cycling Conjecture holds for the e-paths of even-cycle matroids.…”

“…In a flow problem we are given a graph G where the edges are partitioned into demand edges Σ and capacity edges E(G) − Σ. 2 Every edge e is assigned a non-negative integer value w e . For a demand edge e, w e indicates the amount of flow required between the endpoints of e, and for a capacity edge e, w e is the maximum amount of flow allowed on that edge.…”

“…In the next proposition [2] we indicate a number of classes of signed grafts that do not contain odd-K 5 and do not contain L 7 . In particular, for each of these classes, as long as the Eulerian condition holds, the minimum number of edges needed to intersect all odd T -joins is equal to the maximum number of pairwise disjoint odd T -joins.…”