| Abstract: | An r-graph is any graph that can be obtained as a conic combination of its own 1-factors. An r-graph G(V, E) is said to be indecomposable when its edge set E cannot be partitioned as E = E1 ∪ E2 so that Gi(V, Ei) is an ri-graph for i = 1, 2 and, for some r1, r2. We give an indecomposable r-graph for every integer r ≥ 4. This answers a question raised in Seymour, Proc London Math Soc 38 (1979, 423–460], and has interesting consequences for the Schrijver System of the T-cut polyhedron to be given in Rizzi, 1997, to appear]. A graph in which every two 1-factors intersect is said to be poorly matchable. Every poorly matchable r-graph is indecomposable. We show that for every r ≥ 4 that “being indecomposable” does not imply “being poorly matchable.” Next we give a poorly matchable r-graph for every r ≥ 4. The article provides counterexamples to some conjectures of Seymour. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 1–15, 1999 |
