The Kromatic symmetric function: A K -theoretic analog of $X_G$

Abstract Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with K -theory yields a rich combinatorial theory of inhomogeneous deformations, where Schur functions are replaced by their K -analogs, the symmetric Grothendieck functions . We initiate a theory of the Kromatic symmetric function $\overline {X}_G$ , a K -theoretic analog of the chromatic symmetric function $X_G$ of a graph G . The Kromatic symmetric function is a generating series for graph colorings in which vertices receive any nonempty set of colors such that neighboring color sets are disjoint. Our main result lifts a theorem of Gasharov (1996), showing that when G is a claw-free incomparability graph, $\overline {X}_G$ is a positive sum of symmetric Grothendieck functions. This suggests a topological interpretation of Gasharov’s theorem. Kromatic symmetric functions of path graphs are not positive in any of several K -analogs of the e -basis, demonstrating that the Stanley–Stembridge conjecture (1993) does not have such a lift to K -theory and so is unlikely to be amenable to a topological perspective. We define a vertex-weighted extension of $\overline {X}_G$ which admits a deletion–contraction relation. Finally, we give a K -analog for $\overline {X}_G$ of the monomial-basis expansion of $X_G$ .