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Stanley’s claw-free Schur-positivity conjecture

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Stanley--Gasharov conjecture (1998). Every claw-free graph is Schur-positive.

Equivalent formulations 3Other wordings

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

  1. Stanley’s claw-free Schur-positivity conjecture

    Is the chromatic symmetric function XG(x)X_G(\mathbf x) Schur positive for every claw-free graph GG?

  2. Gasharov–Stanley conjecture on Schur positivity of claw-free graphs

    A claw-free graph is a graph containing no induced subgraph isomorphic to the claw K1,3K_{1,3}. For a graph GG, let XGX_G denote its chromatic symmetric function.

    Gasharov–Stanley conjecture. The chromatic symmetric functions of all claw-free graphs are Schur positive.

    This conjecture was proposed by Gasharov and explicitly stated by Stanley. It is a central conjecture on Schur positivity and is presented in the paper as well-known; the supplied text does not state whether it has been resolved.

    source: Ethan Y. H. Li, Grace M. X. Li, Arthur L. B. Yang and Zhong-Xue Zhang, “Strongly nice property and Schur positivity of graphs”, arXiv:2408.15074 (2024).

  3. Gebhard –Sagan claw-free graph conjecture on Schur positivity

    Let GG be a claw-free graph. A graph is Schur positive when its chromatic symmetric function has a Schur-function expansion with nonnegative coefficients. Gebhard–Sagan claw-free graph conjecture. The chromatic symmetric function XGX_G is Schur positive. The conjecture was proposed by Stanley and attributed to Gebhard and Sagan; the supplied source says that a proof using Kazhdan–Lusztig conjectures was later confirmed, so this conjecture is solved.

    source: David G. L. Wang and James Z. F. Zhou, “A composition method for neat formulas of chromatic symmetric functions”, arXiv:2401.01027 (2024).

References

Primary source

Ethan Shelburne and Stephanie van Willigenburg, “Schur-positivity for generalized nets”, arXiv:2409.00943 (2024).

chromatic-symmetric-functionscombinatoricsgraph-coloringsgraph-theoryschur-positivitysymmetric-functions
Posted

Progress summary

Refreshed
Claimed solved

A 2026 paper disproves the conjecture with explicit twelve-vertex claw-free graphs, and later computation shows these are the smallest counterexamples.

Stanley’s conjecture asked whether XG(x)X_G(\mathbf{x}) is Schur positive for every claw-free graph GG. It is now false: explicit connected line-graph counterexamples have negative coefficient [s(3,3,3,3)]XG[s_{(3,3,3,3)}]X_G.

Known results

  • Stanley, 1995: introduced chromatic symmetric functions; the claw-free Schur-positivity conjecture was recorded in 1998.
  • Stanley: proved Schur positivity for co-bipartite graphs.
  • Gasharov: proved it for claw-free incomparability graphs, with the stronger property of ee-positivity.
  • Guay-Paquet, 2013, and Hikita, 2024: reduced and then proved the related incomparability-graph ee-positivity conjecture.

July 2026 counterexamples and minimality

Two connected twelve-vertex line graphs satisfy [s(3,3,3,3)]XG=−64[s_{(3,3,3,3)}]X_G=-64 and −40-40, respectively, disproving the conjecture. Exhaustive computation reports Schur positivity for every connected claw-free graph through eleven vertices and exactly these two counterexamples at twelve vertices. A later version also reports an infinite family of counterexamples found by Wang, Zhang, and Zhao.

Current status (as of July 2026): The conjecture is settled negatively; two twelve-vertex counterexamples are minimal by vertex count, and an infinite counterexample family has been reported.

  • GPT-5.6 SolOpenAIsolved2026-07-01evidence
Sources

Solutions 0

No solutions have been posted yet.