Indecomposable r-graphs and some other counterexamples

Rizzi, Romeo (1999) Indecomposable r-graphs and some other counterexamples. UNSPECIFIED.

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    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 indecomposable when its edge set E cannot be partitioned as the disjoint union of sets E_1 and E_2 so that (V,E_i) is an r_i-graph for i=1,2 and for some r_1, r_2. We give an indecomposable r-graph for every integer r >= 4. This answers a question raised in 1: [ P.D. Seymour, On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte. Proceedings of the London Mathematical Society, Vol. 38, 423--460, (1979)] and 2: [ P.D. Seymour, Some unsolved problems on one-factorizations of graphs. Graph Theory and Related Topics, J.A. Bondy and U.S.R. Murty, Eds., 367--368, Academic Press, New York, 1979] and has interesting consequences for the Schrijver System of the T-cut polyhedron to be given in 3: [ R. Rizzi, Minimum T-cuts and optimal T-pairings, Discrete Mathematics, 2002]. 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 "being indecomposable" does not imply "being poorly matchable". Next we give a poorly matchable r-graph for every r >= 4. The paper provides counterexamples to some conjectures of Seymour first appeared in [1] and [2].

    Item Type: Departmental Technical Report
    Department or Research center: Information Engineering and Computer Science
    Subjects: Q Science > QA Mathematics > QA075 Electronic computers. Computer science
    Uncontrolled Keywords: r-graph; indecomposable; Petersen graph; Fulkerson Coloring
    Additional Information: Published in: "Journal of Graph Theory" 32 (1) (1999) 1--15.
    Report Number: DIT-02-052
    Repository staff approval on: 21 Jan 2003

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