206 research outputs found
The Brown-Colbourn conjecture on zeros of reliability polynomials is false
We give counterexamples to the Brown-Colbourn conjecture on reliability
polynomials, in both its univariate and multivariate forms. The multivariate
Brown-Colbourn conjecture is false already for the complete graph K_4. The
univariate Brown-Colbourn conjecture is false for certain simple planar graphs
obtained from K_4 by parallel and series expansion of edges. We show, in fact,
that a graph has the multivariate Brown-Colbourn property if and only if it is
series-parallel.Comment: LaTeX2e, 17 pages. Version 2 makes a few small improvements in the
exposition. To appear in Journal of Combinatorial Theory
On two conjectures of Maurer concerning basis graphs of matroids
We characterize 2-dimensional complexes associated canonically with basis
graphs of matroids as simply connected triangle-square complexes satisfying
some local conditions. This proves a version of a (disproved) conjecture by
Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14
(1973), 216-240). We also establish Conjecture 1 from the same paper about the
redundancy of the conditions in the characterization of basis graphs. We
indicate positive-curvature-like aspects of the local properties of the studied
complexes. We characterize similarly the corresponding 2-dimensional complexes
of even -matroids.Comment: 28 page
Determining a binary matroid from its small circuits
It is well known that a rank-r matroid M is uniquely determined by its circuits of size at most r. This paper proves that if M is binary and r ≥ 3, then M is uniquely determined by its circuits of size at most r - 1 unless M is a binary spike or a special restriction thereof. In the exceptional cases, M is determined up to isomorphism
Negative correlation and log-concavity
We give counterexamples and a few positive results related to several
conjectures of R. Pemantle and D. Wagner concerning negative correlation and
log-concavity properties for probability measures and relations between them.
Most of the negative results have also been obtained, independently but
somewhat earlier, by Borcea et al. We also give short proofs of a pair of
results due to Pemantle and Borcea et al.; prove that "almost exchangeable"
measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious"
example of a class of measures for which this important property can be shown
to hold; and mention some further questions.Comment: 21 pages; only minor changes since previous version; accepted for
publication in Random Structures and Algorithm
- …