Distinguish all graphs by the Kromatic symmetric function

Determine whether the Kromatic symmetric function ${X}_G$ distinguishes every pair of nonisomorphic graphs.

Background

The paper establishes that the Kromatic symmetric function XG{X}_G can distinguish some graphs that the ordinary chromatic symmetric function cannot. It remains unresolved whether XG{X}_G distinguishes all graphs. The paper also records a conjecture from earlier work asserting that it does, supported by results showing that XG{X}_G determines the multiset of independence polynomials of induced subgraphs and thereby recovers counts of many classes of induced subgraphs.

References

${X}_G$ contains more information than $X_G$ and is known to distinguish some graphs that $X_G$ cannot , but it is unknown whether it distinguishes all graphs or even all trees. In , we conjectured that ${X}_G$ does distinguish all graphs and gave evidence toward that conjecture by showing that the number of copies in $G$ of certain induced subgraphs can be recovered from ${X}_G$, by setting up large systems of linear equations based on the ${m}$-expansion of ${X}_G$ from .

— Power sum expansions for Kromatic symmetric functions using Lyndon heaps  (2502.21285 - Pierson, 28 Feb 2025) in Section 1, subsection “Counting induced subgraphs using ${X}_G$”