Clocks are e-positive

Along with his confirmation of the e -positivity of all cycle-chord graphs θ a b 1 , the third author conjectured the e -positivity of all theta graphs θ a b c . In this paper, we establish the e -positivity of all clock graphs θ a b 2 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.