On Packing T-Joins

A graft is a graph G = (V,E) together with a set T ⊆ V of even cardinality. A T-cut of G is an edge cut δ(X) for which |X ∩T | is odd. A T-join of G is a set of edges S ⊆ E with the property that a vertex of the graph (V, S) has odd degree if and only if it is in T . A T-join packing of G is a set of pairwise disjoint T-joins. Let τ(G) be the size of the smallest T-cut of G and let ν(G) be the size of the largest T-join packing of G. It is an easy fact that every T-cut and every T-join intersect. Thus, ν(G) ≤ τ(G). In this paper, we prove that ν(G) ≥ b6τ(G)c. In the specific case that G is eulerian, or T = {v ∈ V | deg(v) is odd}, we prove that ν(G) ≥ b1 2τ(G)c. This resolves conjecture of Zhang.