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.
Sublinear separators, fragility and subexponential expansion.European Journal of Combinatorics, 52:103–119, 2016
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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.