Let Delta_m,n be the chessboard complex, whose faces are the matchings in the complete bipartite graph K_m,n, with n≥ m≥ 1. Let S=k[x_ij:1≤ i≤ m,,1≤ j≤ n], and let I_Delta_m,n be the facet ideal generated by ∏_(i,j)∈ Fx_ij as F ranges over the facets of Delta_m,n. The conjecture asserts that operatornamedepth(S/I_Delta_m,n)=operatornamereg(S/I_Delta_m,n)=2(m-1) for every field k and all n≥ m≥ 1.

Conjecture on facet ideals of chessboard complexes

Let $\Delta_{m,n}$ be the chessboard complex, whose faces are the matchings in the complete bipartite graph $K_{m,n}$ , with $n\geq m\geq 1$ . Let $S=k[x_{ij}:1\leq i\leq m,\,1\leq j\leq n]$ , and let $I_{\Delta_{m,n}}$ be the facet ideal generated by $\prod_{(i,j)\in F}x_{ij}$ as $F$ ranges over the facets of $\Delta_{m,n}$ . The conjecture asserts that $\operatorname{depth}(S/I_{\Delta_{m,n}})=\operatorname{reg}(S/I_{\Delta_{m,n}})=2(m-1)$ for every field $k$ and all $n\geq m\geq 1$ .

References

Primary source

arXiv

Additional references

Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A September 2026 preprint claims to settle the conjecture, but its broadest conclusions are conditional and the result has not been independently verified.

The 2022 paper formulates the conjecture that for the chessboard complex with $n \ge m \ge 1$ , both depth and regularity equal $2(m-1)$ .

Known results

The conjecture was proved for $m \le 3$ in the 2022 paper.
The case $(m,n)=(4,4)$ was checked computationally, yielding depth and regularity $6$ .
The same paper established lower bounds $2(m-1)$ in all dimensions and exact power formulas for $m \le 2$ .

September 2, 2026 claimed resolution

The preprint Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals claims homology injections under edge addition and corresponding Leray, regularity, depth, projective-dimension, maximal-shift, and extremal-Betti-number results. It claims to resolve the conjecture, with complete invariant calculations for $K_{r,s}$ ; the broader graph-level conclusions require sufficiently long cycles. The claim is unverified.

Current status (as of September 2026): The conjecture is claimed solved by the 2026 preprint, but verification is absent; its fully explicit invariant determination is established only for $K_{r,s}$ , while broader conclusions are conditional.

Sources

No solutions have been posted yet.