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arXiv · 2607.26364

A counterexample to the claw-free Schur-positivity conjecture

Jitendra Prajapati
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Abstract

The claw-free Schur-positivity conjecture, recorded by Stanley (1998) and credited there to Gasharov, asserts that the chromatic symmetric function of every claw-free graph is Schur-positive. We give a counterexample on 12 vertices: the line graph $G$ of the graph obtained from a 4-cycle by attaching triangles at two opposite vertices and pendant edges at the other two satisfies $[s_{(3,3,3,3)}]X_G = -64$. The coefficient follows from a short computation by hand and is also reproduced by three exact implementations. An exhaustive computation over all 216,777 connected claw-free graphs on at most 11 vertices shows that every one is Schur-positive, so 12 vertices is the minimum order of any counterexample. A complete census of the 1,728,404 connected claw-free graphs on 12 vertices finds exactly two non-Schur-positive isomorphism classes; the other has graph6 code K?`CR@`bAbRB and coefficient $[s_{(3,3,3,3)}] = -40$.

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Jitendra Prajapati. 2026-07-29. A counterexample to the claw-free Schur-positivity conjecture. https://arxiv.org/abs/2607.26364

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