For every fixed integer n≥ 2, the family of undirected generalized pancake graphs P(m,n)_m≥ 2 is an expander family; equivalently, there exists a constant c_n>0 such that gamma(P(m,n))≥ c_n for every m≥ 2, where gamma(P(m,n)) denotes the normalized Laplacian spectral gap.

Blanco–Buehrle generalized pancake expander conjecture

Blanco and Buehrle conjectured that, for each fixed $n$ , generalized pancake graphs form an expander family as the colour parameter grows. A new preprint by Saúl A. Blanco claims this is false for fixed $n>2$ .

Known results

June 2025: the expander conjecture was described as open for $m>1$ , with upper bounds known for the spectral gap.
September 2025: a note proved other Blanco-Buehrle spectral-gap and multiplicity conjectures, not the expander assertion.

August 2026 claimed disproof

The preprint proves, for $m,n\ge 2$ , bounds of order $1/n$ , and for fixed $n\ge 2$ bounds implying $\gamma(P(m,n))=\Theta_n(m^{-2})$ as $m\to\infty$ . It therefore claims that $\{P(m,n)\}_{m>2}$ is not an expander family for fixed $n>2$ .

Current status (as of August 2026): The expander conjecture is claimed disproved by an unrefereed preprint; its asymptotic bounds are not yet independently verified, and general closed spectral-gap formulas remain open.

Blanco–Buehrle generalized pancake expander conjecture

Progress summary

Primary source

Additional references

Solutions 0