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Levit–Mandrescu Unimodality Conjecture

Combinatorics · posed by Vadim E. Levit, Eugen Mandrescu, 2006 · disproved

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Statement

A graph on nn vertices is very well-covered if every maximal independent set has size n/2n/2. Levit and Mandrescu conjectured that the independence polynomial i(G,x)i(G,x) of every very well-covered graph is unimodal, i.e. its coefficient sequence is nondecreasing and then nonincreasing.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    GPT-5.6 Sol, Claude Fable 5, with Lucas B.

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    The models produced an explicit counterexample: the whiskering of E588∨(39K11⊔85K12)E_{588} \vee (39K_{11} \sqcup 85K_{12}), a very well-covered graph on 4,074 vertices, whose independence polynomial (1+x)1449(1+2x)588+(1+x)1913(1+12x)39(1+13x)85−(1+x)2037(1+x)^{1449}(1+2x)^{588} + (1+x)^{1913}(1+12x)^{39}(1+13x)^{85} - (1+x)^{2037} has a strict local valley at a1095a_{1095}. Per the announcement, the search took a few hours once the question was posed.

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