Madani–Kiani's maximal-clique regularity conjecture for binomial edge ideals

Madani–Kiani conjectured that the regularity of a graph's binomial edge ideal is at most its number of maximal cliques. The conjecture was originally open in general, despite many established special cases.

Known results

• Closed graphs: $\operatorname{reg}(S/J_G)\leq c(G)$ (Ene, Herzog, Hibi, 2012).
• Chordal graphs: the conjectured bound holds (Ene, Herzog, Hibi, 2018).
• Further special classes include generalized block, fan, $P_4$-free, block, and semi-block graphs.

All-graph proof and 2022 confirmation

Malayeri, Saeedi Madani, and Kiani claim an all-graph proof (publication date not stated in the scan): they establish the sharper inequality $\operatorname{reg}(S/J_G)\leq\eta(G)$, where $\eta(G)\leq c(G)$, implying the conjecture. A 2022 survey presents this result as resolving the conjecture. No retrieved source reports a gap, retraction, or independent verification.

Current status (as of September 2026): An all-graph proof is claimed, with the sharper bound $\operatorname{reg}(S/J_G)\leq\eta(G)\leq c(G)$, but the resolution remains unverified in this report.