Packing Non-Zero A-Paths In Group-Labelled Graphs

Authors:
Maria Chudnovsky;Jim Geelen;Bert Gerards;Luis Goddyn;Michael Lohman;Paul Seymour
Affiliations:
Princeton University, Department of Mathematics, Princeton NJ08544, USA;University of Waterloo, Department of Combinatorics and Optimization, Waterloo, N2L 3G1, Canada;CWI Postbus 94079, 1090 GB, Amsterdam, The Netherlands and Eindhoven University of Technology, Postbus 513, Department of Mathematics and Computer Science, 600 MB, Eindhoven, The Netherlands;Simon Fraser University, Department of Mathematics, 1090 GB, Burnaby V5A 1S6, Canada;Princeton University, Department of Mathematics, 1090 GB, Princeton NJ08544, USA;Princeton University, Department of Mathematics, 1090 GB, Princeton NJ08544, USA
Venue:
Combinatorica
Year:
2006

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Abstract

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 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.