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Indecomposable r ‐graphs and some other counterexamples
Author(s) -
Rizzi Romeo
Publication year - 1999
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/(sici)1097-0118(199909)32:1<1::aid-jgt1>3.0.co;2-b
Subject(s) - indecomposable module , combinatorics , counterexample , mathematics , graph , discrete mathematics
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 = E 1 ∪ E 2 so that G i ( 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 [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