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. (C) 2004 Elsevier Inc. All rights reserved.