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

Gordon Royle, Alan D. Sokal

    Research output: Contribution to journalArticle

    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.

    Original languageEnglish (US)
    Pages (from-to)345-360
    Number of pages16
    JournalJournal of Combinatorial Theory, Series B
    Volume91
    Issue number2
    DOIs
    StatePublished - Jul 2004

    Fingerprint

    Polynomials
    Polynomial
    Univariate
    Zero
    Simple Graph
    Series Expansion
    Complete Graph
    Planar graph
    Counterexample
    If and only if
    Series
    Graph in graph theory
    False
    Form

    Keywords

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

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Theoretical Computer Science

    Cite this

    The Brown-Colbourn conjecture on zeros of reliability polynomials is false. / Royle, Gordon; Sokal, Alan D.

    In: Journal of Combinatorial Theory, Series B, Vol. 91, No. 2, 07.2004, p. 345-360.

    Research output: Contribution to journalArticle

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