Clocks are $e$-positive

L. Chen, Y. T. He, David G. L. Wang

Published 2024-10-10Version 1

Along with his confirmation of the $e$-positivity of all cycle-chord graphs $\theta_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $\theta_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $\theta_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least $2$.