  The machinery of noncommutative Schur functions is a general approach to
Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang
recently adapted this theory to posets to give a new approach to the
Stanley-Stembridge conjecture. We further develop this theory to prove that the
symmetric function associated to any $P$-Knuth equivalence graph is Schur
positive. This settles a conjecture of Kim and the third author, and refines
results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of
chromatic symmetric functions.
