Hopf algebra maps taking chromatic symmetric functions to their graph complements

Cho and van Willigenburg (arXiv:1508.07670) and Alinaeifard, Wang, and van Willgenburg (arXiv:2010.00147) introduce multiplicative chromatic bases for the ring $\Lambda$ of symmetric functions, consisting of the chromatic symmetric functions (CSFs) of a sequence of connected graphs $G_1,G_2,\dots$ such that $G_n$ has total weight $n$, together with the CSFs of their disjoint unions. In arXiv:1707.04058, Tsujie introduces an alternative ring structure $\widetilde{\Lambda}$ on the vector space $\Lambda$ that makes CSFs multiply over joins instead of over disjoint unions. The $\widetilde{m}_\lambda$ basis, consisting of all CSFs of weighted cliques, is a multiplicative basis for $\widetilde{\Lambda}$, as is the $r_\lambda$ basis of complete multipartite graphs studied by Penaguiao (arXiv:1803.08824) and Crew and Spirkl (arXiv:2009.14141). We show that one can get more of these "cochromatic bases" (where the starting graphs are combined by joins instead of disjoint unions, hence forming a multiplicative basis for $\widetilde{\Lambda}$ instead of $\Lambda$) if and only if the starting graphs are edgeless. We also show that $\widetilde{\Lambda}$ is a Hopf algebra with the same coproduct as $\Lambda$, and that many of the chromatic bases for $\Lambda$ generated by cliques can be taken to their corresponding cochromatic bases via Hopf algebra isomorphisms $\Lambda \to \widetilde{\Lambda}.$ We also show that there is a single Hopf algebra morphism taking the CSFs of all unweighted triangle-free graphs to the CSFs of their complements, and we give several more conditions and examples for when one can or cannot find Hopf algebra maps taking the CSFs of certain graphs to the CSFs of their complements. Finally, we show that $K$-analogues of many of the above statements also hold if one instead uses the Kromatic symmetric function (KSF) defined by Crew, Pechenik, and Spirkl (arXiv:2301.02177).