7.
Let
G=(
V,E) be an oriented graph whose edges are labelled by the elements of a group
Γ and let
A⊂V. An
A-path is a path whose ends are both in
A. The
weight of a path
P in
G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in
P. (If
Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint
A-paths each of non-zero weight. When
A =
V this problem is equivalent to the maximum matching problem. The general case also includes Mader's
S-paths problem. We prove that for any positive integer
k, either there are
k vertex-disjoint
A-paths each of non-zero weight, or there is a set of at most 2
k −2 vertices that meets each of the non-zero
A-paths. This result is obtained as a consequence of an exact min-max theorem.
These results were obtained at a workshop on Structural Graph Theory at the PIMS Institute in Vancouver, Canada. This research
was partially conducted during the period the first author served as a Clay Mathematics Institute Long-Term Prize Fellow.
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