AI 中文总结
该研究针对Stanley记载的无爪图Schur正性猜想，构造出12顶点的反例，经计算验证其色对称函数系数为负，且12顶点是最小反例阶数，还发现另一个同构类反例。
AI 中文摘要
Stanley（1998）记载的、归功于Gasharov的无爪图Schur正性猜想断言：每个无爪图的色对称函数都是Schur正的。我们给出一个12顶点的反例：由4-圈在两个相对顶点处附加三角形、另外两个顶点处附加悬挂边得到的图的线图$G$满足$[s_{(3,3,3,3)}]X_G=-64$。该系数通过手动短计算得到，也被三个精确实现复现。对所有216777个顶点数不超过11的连通无爪图的穷尽计算显示，每个都是Schur正的，因此12顶点是任何反例的最小阶数。对12顶点的1728404个连通无爪图的完整普查发现恰好有两个非Schur正的同构类；另一个的graph6编码为K?`CR@`bAbRB，系数$[s_{(3,3,3,3)}]=-40$。
英文摘要
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 $[s_{(3,3,3,3)}] = -40$.
Comments4 pages. Verification code and exhaustive census data at https://github.com/infinityscroll/claw-free-schur-counterexample
