Abstract
Motivated by Stanley and Stembridge's $(\mathbf{3}+\mathbf{1})$-free conjecture on chromatic symmetric functions, Foley, Hoàng 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 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, Hoàng and Merkel conjectured that if $G$ is $e$-positive, then so is the resulting twin graph $G_v$ for any vertex $v$. By considering the twinning operation on a subclass of tadpole graphs with respect to certain vertices we disprove the latter conjecture. We further show that if $G$ is $e$-positive, the twin graph $G_v$ and more generally the clan graphs $G^{(k)}_v$ ($k \geq 1$) may not even be $s$-positive, where $G^{(k)}_v$ is obtained from $G$ by applying $k$ twinning operations to $v$.
Funding Statement
The third author is supported by the National Science
Foundation of China (No. 11671037). The fourth author is supported in part by the Fundamental
Research Funds for the Central Universities and the National Science Foundation of China
(Nos. 11522110 and 11971249).
Citation
Ethan Y. H. Li. Grace M. X. Li. David G. L. Wang. Arthur L. B. Yang. "The Twinning Operation on Graphs Does not Always Preserve $e$-positivity." Taiwanese J. Math. 25 (6) 1089 - 1111, December, 2021. https://doi.org/10.11650/tjm/210703
Information