For every integer d≥2, the paper constructs graphs of degree-d polynomial growth whose balanced separators are too large by a logarithmic factor to fit in the conjectured product structure.
Asymptotic dimension of minor-closed families and Assouad-Nagata dimension of surfaces.Journal of the European Mathematical Society, 26(10):3739–3791, 2024
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Subdivided expanders and counterexamples to the Tree Product Conjecture
For every integer d≥2, the paper constructs graphs of degree-d polynomial growth whose balanced separators are too large by a logarithmic factor to fit in the conjectured product structure.