Saturated Newton polytopes of Schur-positive chromatic symmetric functions
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Does Schur positivity of a graph’s chromatic symmetric function imply a saturated Newton polytope after every finite-variable specialization? Schur positivity means its Schur-basis coefficients are nonnegative. Saturation means that every lattice point in the convex hull of the monomial exponent vectors actually occurs in the polynomial.
Why it matters and what counts as progress
Why it matters: Monical’s conjecture links positivity in one basis to the absence of missing lattice-point weights in another description of graph-coloring polynomials.
What counts as progress: Prove the implication for a new graph family or find a graph with certified Schur positivity and a missing lattice-point weight. Both parts of a counterexample must concern the same graph and specialization; an unclear positivity certificate does not establish the refutation.
How the problem developed
- 23 Jul 2026 · Counterexample [1]
Emilie Matherne and Alejandro H. Morales Gave a 12-vertex, 19-edge bipartite graph with a reported Schur-positive chromatic symmetric function whose three-variable specialization omits a lattice point in its Newton polytope.
AI contribution
Chromatic symmetric functions of claw-free graphs are not Schur positive
Matherne and Morales give a 12-vertex, 19-edge bipartite graph G3 whose chromatic symmetric function is reported Schur positive while its three-variable specialization omits lattice-point weight (7,4,1) between present weights (6,6,0) and (8,2,2).
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