The Twinning Operation on Graphs Does not Always Preserve e-positivity

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 υ, which adds a vertex υ′ to G such that υ and υ′ 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 Gυ for any vertex υ. 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 Gυ 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 υ.