Kashima’s pancyclicity analogue conjecture

Kashima’s conjecture predicts a minimum-degree threshold of order $\sqrt{3n}$ for the relevant pancyclicity analogue. The conjecture itself is not settled.

August 2026 sublinear bound

A preprint titled “A Sublinear Minimum-Degree Condition for 2-Connected Subgraphs of All Orders” reduces the sufficient minimum-degree requirement from linear order to a sublinear bound. This is the first such sublinear sufficient condition toward Kashima’s conjectured $\sqrt{3n}$ scale, but it does not establish that threshold.

Current status (as of August 2026): A sublinear sufficient condition is available, while Kashima’s conjectured $\sqrt{3n}$ threshold and the full conjecture remain open.