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

Gordon Royle, Alan D. Sokal

    Research output: Contribution to journalArticlepeer-review


    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.

    Original languageEnglish (US)
    Pages (from-to)345-360
    Number of pages16
    JournalJournal of Combinatorial Theory. Series B
    Issue number2
    StatePublished - Jul 2004


    • All-terminal reliability
    • Brown-Colbourn conjecture
    • Potts model
    • Reliability polynomial
    • Tutte polynomial

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Discrete Mathematics and Combinatorics
    • Computational Theory and Mathematics


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