Counting induced subgraphs with the Kromatic symmetric function

The chromatic symmetric function XG is a sum of monomials corresponding to proper vertex colorings of a graph G. Crew, Pechenik, and Spirkl (2023) recently introduced a K-theoretic analogue X̄G called the Kromatic symmetric function, where each vertex is instead assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. XG does not distinguish all graphs, but a longstanding open question is whether it distinguishes all trees. We conjecture that X̄G does distinguish all graphs. As evidence towards this conjecture, we show that X̄G determines the number of copies in G of certain induced subgraphs on 4 and 5 vertices as well as the number of induced subgraphs isomorphic to each graph consisting of a star plus some number of isolated vertices.