## Abstract

Motivated by Stanley and Stembridge's (3+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_{v} ^{(k)} (k ≥ 1) may not even be s-positive, where G_{v} ^{(k)} is obtained from G by applying k twinning operations to v.

Original language | English |
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Pages (from-to) | 1089-1111 |

Number of pages | 23 |

Journal | Taiwanese Journal of Mathematics |

Volume | 25 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2021 |

## Keywords

- Chromatic symmetric functions
- Chromatic symmetric functions in non-commuting variables
- E-positivity
- S-positivity
- Tadpole graphs
- Twinning operation