Packing non-zero AA-paths in group-labelled graphs

Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group G 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 G 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 2k -2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem

Smallest C2l+1C_{2l+1}-Critical Graphs of Odd-Girth 2k+12k+1

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