AI 中文总结
该研究通过结合无爪图的阶数普查数据，证明了Stanley–Gasharov猜想的最小反例，并构造出无穷族连通线图作为该猜想的无穷反例。
AI 中文摘要
Stanley–Gasharov猜想断言，每个无爪图的色对称函数都是Schur-正的。Prajapati以及Matherne和Morales分别找到了反例，其中后者要求构造无穷反例族。结合Prajapati对阶数≤12的完整普查，以及对n个顶点、m条边（13≤n≤21，n-1≤m≤20）的144492个此前未处理的连通无爪图的精确普查，我们表明其具有12个顶点、21条边的反例图G₂是按边优先、顶点次优先序下的唯一最小反例。我们还构造了无穷族连通线图，其色对称函数不是Schur-正的，这就得到了Stanley–Gasharov猜想的无穷反例族。
英文摘要
The Stanley--Gasharov conjecture asserts that every claw-free graph is Schur-positive. Prajapati and, independently, Matherne and Morales identified the same pair of counterexamples, both of which are line graphs, thereby disproving the conjecture. In this paper, we construct two infinite families of counterexamples to the Stanley--Gasharov conjecture, thereby answering a question of Matherne and Morales. Every graph in the first family is a line graph, whereas no graph in the second family is a line graph. Prajapati further showed that the graph $G_2$, which has $12$ vertices and $21$ edges, is the smallest counterexample under the ordering that first compares the numbers of vertices and then the numbers of edges. We show that $G_2$ is also the smallest counterexample under the reverse ordering, which first compares the edge numbers and then the vertex numbers. Similarly, we exhibit a graph $Q$ with $13$ vertices and $27$ edges and show that $Q$ is the smallest counterexample that is not a line graph under each ordering. Our two infinite families are obtained from $G_2$ and $Q$, respectively, by adjoining a clique of order at least $4$ and connecting one of its vertices to a distinguished vertex of the original graph by a single edge.
Comments12 pages, 7 figures
