Let X_G be the chromatic symmetric function of a graph G, and let X_G(x_1,…,x_k) denote its specialization to k variables. A polynomial is SNP (has a saturated Newton polytope) if its support equals the set of lattice points in its Newton polytope. Monical's conjecture. If X_G is s-positive, then X_G(x_1,…,x_k) is SNP for any k. This conjecture concerns the relationship between Schur positivity and saturated Newton polytopes for finite-variable chromatic symmetric polynomials; the paper gives a counterexample, so the asserted implication does not hold in general.

Monical’s SNP conjecture for Schur-positive chromatic functions

About 8 years old · traced to 2018

Let $X_G$ be the chromatic symmetric function of a graph $G$ , and let $X_G(x_1,\ldots,x_k)$ denote its specialization to $k$ variables. A polynomial is SNP (has a saturated Newton polytope) if its support equals the set of lattice points in its Newton polytope. Monical's conjecture. If $X_G$ is $s$ -positive, then $X_G(x_1,\ldots,x_k)$ is SNP for any $k$ . This conjecture concerns the relationship between Schur positivity and saturated Newton polytopes for finite-variable chromatic symmetric polynomials; the paper gives a counterexample, so the asserted implication does not hold in general.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Monical’s SNP conjecture for Schur-positive chromatic functions
If $X_G$ is Schur positive, must $X_G(x_1,\ldots,x_k)$ have saturated Newton polytope for every finite $k$ ?

References

Primary source

Jacob P. Matherne and Alejandro H. Morales, “Chromatic symmetric functions of claw-free graphs are not Schur positive”, arXiv:2607.21508 (2026).

Additional references

3 papers in this index state this conjecture (2018–2026). The statement above is taken from the most recent of them; the others are arXiv:2201.07333, arXiv:1810.10361.

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 paper gives a concrete counterexample, so Schur positivity does not always guarantee the required no-gaps property.

Monical’s conjecture asserted that every Schur-positive chromatic symmetric function has a saturated Newton polytope after restricting to finitely many variables. The assertion is now disproved by an explicit bipartite graph.

Known results

Co-bipartite graphs, indifference graphs of Dyck paths, and incomparability graphs of $(3+1)$ -free posets satisfy SNP for every finite $k$ ; their Newton polytopes are permutahedra (2022).

July 2026 counterexample

The graph $G_3$ has $12$ vertices, and $X_{G_3}$ is Schur positive but not SNP. In three variables, weights $(6,6,0)$ and $(8,2,2)$ occur, while their midpoint $(7,4,1)$ does not: $[m_{741}]X_{G_3}(x_1,x_2,x_3)=0$ . Thus the universal implication is false. The examples were reportedly found using ChatGPT-5.6 Sol Pro, but the source does not specify its precise mathematical contribution.

Current status (as of July 2026): The conjecture is disproved by $G_3$ ; only restricted graph classes remain covered by the positive results.

GPT-5.6 SolOpenAIsolved2026-07-01evidence

Sources

No solutions have been posted yet.