标题： 图上的孪生操作并不总是保持$e$-正性 显示英文标题

标题： The twinning operation on graphs does not always preserve $e$-positivity

摘要： 受Stanley关于色对称函数的$\mathbf{(3+1)}$-自由猜想的启发，Foley、Hoàng和Merkel引入了强$e$-正性的概念，并猜想一个图是强$e$-正性当且仅当它是（爪，网）-自由的。 为了研究强$e$-正图，他们进一步引入了在图$G$上相对于顶点$v$的孪生操作，该操作向$G$添加一个顶点$v'$，使得$v$和$v'$相邻，并且任何其他顶点都与它们两者相邻或都不相邻。 Foley、Hoàng 和 Merkel 猜测，如果$G$是$e$正的，那么对于任何顶点$v$，得到的双图$G_v$也是正的。 基于 Gebhard 和 Sagan 开发的非交换变量着色对称函数理论，我们建立了称为藤蔓图的一类图的$e$正性。 通过考虑这些图的一个子类在某些顶点上的双操作，我们驳斥了 Foley、Hoàng 和 Merkel 的后一个猜想。 我们进一步证明，如果$G$是$e$-正的，那么孪生图$G_v$以及更一般地，族图$G^{(k)}_v$($k \ge 1$) 甚至可能不是$s$-正的，其中$G^{(k)}_v$是通过将$k$次双生操作应用于$v$而从$G$得到的。 显示英文摘要

摘要： Motivated by Stanley's $\mathbf{(3+1)}$-free conjecture on chromatic symmetric functions, Foley, Ho\`{a}ng and Merkel introduced the concept of strong $e$-positivity and conjectured that a graph is strongly $e$-positive if and only if it is (claw, net)-free. In order to study strongly $e$-positive graphs, they further introduced the twinning operation on a graph $G$ with respect to a vertex $v$, which adds a vertex $v'$ to $G$ such that $v$ and $v'$ are adjacent and any other vertex is adjacent to both of them or neither of them. Foley, Ho\`{a}ng and Merkel conjectured that if $G$ is $e$-positive, then so is the resulting twin graph $G_v$ for any vertex $v$. Based on the theory of chromatic symmetric functions in non-commuting variables developed by Gebhard and Sagan, we establish the $e$-positivity of a class of graphs called tadpole graphs. By considering the twinning operation on a subclass of these graphs with respect to certain vertices we disprove the latter conjecture of Foley, Ho\`{a}ng and Merkel. We further show that if $G$ is $e$-positive, the twin graph $G_v$ and more generally the clan graphs $G^{(k)}_v$ ($k \ge 1$) may not even be $s$-positive, where $G^{(k)}_v$ is obtained from $G$ by applying $k$ twinning operations to $v$.