分离Schur正性、强良性质与良性质的三个无限族
Three Infinite Families Separating Schur Positivity, the Strongly Nice Property, and the Nice Property
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中文总结 AI 辅助
本文构造三个无限图族，分别分离Schur正性、强良性质与良性质，证明强良图在不交并下封闭，还定义$k$级良性质并确定$N_r$的级深度。
中文摘要 AI 辅助
对于图$G$，$X_G$的Schur正性意味着$G$是强良的，且每个强良图都是良的。我们构造三个无限族来分离这些性质：首先给出连通族$F_t$（$t\ge6$），它是强良的但非Schur正；接着证明具有非负单项式系数的齐次强良对称函数在乘法下封闭，因此强良图在不交并下封闭，作为应用，对$H=K_{3,3}-e$，图$M_t=H\sqcup K_t$（$t\ge3$）构成非连通族，是强良的但非Schur正；最后定义$N_r=K_r\vee(K_2\sqcup2K_1)$（$r\ge2$），证明每个$N_r$是连通且良的但非强良，还引入$k$级良性质，证明$N_r$的级深度为$4r!$。
英文摘要
For a graph $G$, Schur positivity of $X_G$ implies that $G$ is strongly nice, and every strongly nice graph is nice. We construct three infinite families separating these properties. We first give a connected family $F_t$, $t\ge6$, that is strongly nice but not Schur positive. We then prove that homogeneous strongly nice symmetric functions with nonnegative monomial coefficients are closed under multiplication, and hence that strongly nice graphs are closed under disjoint union. As an application, for $H=K_{3,3}-e$, the graphs \[ M_t=H\sqcup K_t,\qquad t\ge3, \] form a disconnected family that is strongly nice but not Schur positive. Finally, we define \[ N_r=K_r\vee(K_2\sqcup2K_1),\qquad r\ge2, \] and prove that every $N_r$ is connected and nice but not strongly nice. We also introduce the level-$k$ nice property and show that the level depth of $N_r$ is $4r!$.
