For every varepsilon>0, there exists a finite simple graph G on n vertices such that delta(G)≥ (3/4-varepsilon)n and G is not globally synchronizing for the Kuramoto model; equivalently, the Kuramoto energy on G has a local minimum other than a fully synchronized state.

Bandeira–Kireeva–Maillard–Rödder Kuramoto conjecture

The conjecture, formalized as Conjecture 5 in BKMR25, asserts that the sharp minimum-degree threshold for global Kuramoto synchronization is $3/4$ : for every $\varepsilon>0$ , some graph with minimum degree at least $(3/4-\varepsilon)n$ is not globally synchronizing.

Known results

Taylor (2011): a $94\%$ minimum-degree condition suffices.
Bandeira, Ling, and Xu (2018): reduced the sufficient threshold to $79.3\%$ .
A 2020 result reduced it to $78.89\%$ .
Kassabov, Strogatz, and Townsend (2021): $3/4$ suffices; lower-bound constructions reached roughly $68\%$ .

August 2026 claimed refutation

An August 2026 arXiv preprint claims that graphs with connectivity $3/4-\varepsilon$ are globally synchronizing, thereby explicitly refuting the Bandeira–Kireeva–Maillard–Rödder conjecture. The claim has not yet been independently verified.

Current status (as of August 2026): The $3/4$ upper bound and earlier lower bounds are established, while the conjecture is challenged by an unrefereed preprint whose claimed refutation remains unverified.

Bandeira–Kireeva–Maillard–Rödder Kuramoto conjecture

Progress summary

Primary source

Additional references

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