Kromatic Quasisymmetric Functions

We provide a construction for the kromatic symmetric function XG of a graph introduced by Crew, Pechenik, and Spirkl using combinatorial (linearly compact) Hopf algebras. As an application, we show that XG has a positive expansion into multifundamental quasisymmetric functions. We also study two related quasisymmetric q-analogues of XG, which are K-theoretic generalizations of the quasisymmetric chromatic function of Shareshian and Wachs. We classify exactly when one of these analogues is symmetric. For the other, we derive a positive expansion into symmetric Grothendieck functions when G is the incomparability graph of a natural unit interval order.