Unicyclic Graphs of Arbitrary Girth with the Same Chromatic Symmetric Function

Abstract

An open question asks whether the chromatic symmetric function (CSF) of a graph distinguishes non-isomorphic trees. While it is known that the CSF does not distinguish unicyclic graphs, examples of pairs of unicyclic graphs with the same CSF and girth larger than 3 were not known until very recently. This manuscript exhibits a sequence of pairs of non-isomorphic, connected, unicyclic graphs with increasing girth and which share the same CSF. This also provides the first infinite family of bipartite graphs with the same CSF. Our main technique is to apply a version of the "triple deletion" modular relation, due independently to Guay-Paquet and Orellana--Scott.