The Brown--Colbourn conjecture on zeros of reliability polynomials is false

Authors:
Gordon Royle;Alan D. Sokal
Affiliations:
Department of Computer Science & Software Engineering, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia;Department of Physics, New York University, 4 Washington Place, New York, NY
Venue:
Journal of Combinatorial Theory Series B
Year:
2004

Citing 6
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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 K4. The univariate Brown-Colbourn conjecture is false for certain simple planar graphs obtained from K4 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.