Stanley’s Schur-positivity conjecture for claw-free graphs

Is the chromatic symmetric function of every claw-free graph Schur-positive, meaning all coefficients in its Schur-function expansion are nonnegative? Stanley’s conjecture includes claw-free graphs beyond incomparability and co-bipartite classes.

Why it matters and what counts as progress

Why it matters: A claw-free counterexample separates Schur positivity from a natural local graph restriction already sufficient in some subclasses.

What counts as progress: Prove positivity for a wider class or give an explicit claw-free graph with a negative Schur coefficient.

How the problem developed

1995 · Conjecture [1]
Richard P. Stanley Introduced the chromatic symmetric function and proposed Schur positivity for claw-free graphs.
23 Jul 2026 · Counterexample [2]
Emilie Matherne and Alejandro H. Morales Gave two 12-vertex claw-free line graphs whose chromatic symmetric functions have negative Schur coefficients −64 and −40.

AI contribution

Chromatic symmetric functions of claw-free graphs are not Schur positive

Jacob P. Matherne, Alejandro H. Morales

Matherne and Morales give two 12-vertex claw-free line graphs whose chromatic symmetric functions have [s₃₃₃₃] coefficients −64 and −40. Either graph refutes Stanley’s claim that every claw-free graph is Schur-positive.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (exact SageMath symbolic computation)

Human understandingFull