Problem detail · source-aware
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A graph on $n$ vertices is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial $i(G,x)$ of every very well-covered graph is unimodal, i.e. its coefficient sequence is nondecreasing and then nonincreasing.
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The models produced an explicit counterexample: the whiskering of $E_{588} \vee (39K_{11} \sqcup 85K_{12})$, a very well-covered graph on 4,074 vertices, whose independence polynomial $(1+x)^{1449}(1+2x)^{588} + (1+x)^{1913}(1+12x)^{39}(1+13x)^{85} - (1+x)^{2037}$ has a strict local valley at $a_{1095}$. Per the announcement, the search took a few hours once the question was posed.
Provider: unknown · Prompt public: unknown · Independence: unknown
Announced on LinkedIn by Lucas B. (Head of AI Research, Jump Trading); no preprint yet, and the reviewers credited are internal to the team rather than independent. The stated polynomial was recomputed with exact integer arithmetic: $a_{1094} > a_{1095} < a_{1096}$ holds, so the polynomial given is genuinely not unimodal, and its low-order coefficients ($a_0 = 1$, $a_1 = 4074$) are consistent with a graph on 4,074 vertices. What remains unchecked is that the whiskered graph's independence polynomial equals the polynomial stated.
Correctness: unknown · statement fidelity: unaudited · peer review: none
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Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.