The Brown-Colbourn conjecture on zeros of reliability polynomials is false
arXiv:math/0301199 · doi:10.1016/j.jctb.2004.03.008
Abstract
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.
LaTeX2e, 17 pages. Version 2 makes a few small improvements in the exposition. To appear in Journal of Combinatorial Theory B