The Cycling Property for the Clutter of Odd st-Walks

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.…”

“…It follows from Theorem 5 that there exists an integer flow y for (G, Σ, 2w). Hence, 1 2 y is a flow for (G, Σ, w). Thus, Theorem 5 implies Theorem 3.…”

“…Here (1) ensures that the total flow carried by a capacity edge e does not exceed its capacity w e , and (2) guarantees that the total flow carried between the endpoints of a demand edge e is equal to its demand w e . Consider the flow instance in Figure 1.…”

“…However, this theorem does not extend to signed matroids. Indeed, we have, τ (L 7 ) = 3 but ν * (L 7 ) = 7 3 (we can assign 1 3 to each of the 7 lines). Hence, L 7 does not fractionally pack.…”