A composition method for neat formulas of chromatic symmetric functions

We develop a composition method to unearth positive $e_I$-expansions of chromatic symmetric functions $X_G$, where the subscript $I$ stands for compositions rather than integer partitions. Using this method, we derive positive and neat $e_I$-expansions for the chromatic symmetric functions of tadpoles, barbells and generalized bulls, and establish the $e$-positivity of hats. We also obtain a compact ribbon Schur analog for the chromatic symmetric function of cycles.