Degree-sequence invariance for chromatic-symmetric-function-equivalent unicyclic graphs

Determine whether there exist two connected unicyclic graphs with a cycle of length greater than three, different degree sequences, and identical chromatic symmetric functions.

Background

The paper presents a conjecture asserting that, for connected unicyclic graphs with either a 3-cycle and an odd number of vertices or a cycle of length at least four, equality of chromatic symmetric functions forces equality of the degree sequence, the number of non-trivial rooted trees, and the number of internal edges. This conjecture had been verified computationally only through 16 vertices.

The authors then isolate the degree-sequence component as a specific unresolved existence question for cycles of length greater than three. The question concerns whether the chromatic symmetric function can fail to distinguish degree sequences in that broader unicyclic setting.