On Calculating the Chromatic Symmetric Function
Nima Amoei Mobaraki, Yasaman Gerivani, Sina Ghasemi Nezhad
TL;DR
The paper develops a unified, graph-structural framework for computing the chromatic symmetric function $X_G(x)$ by contextualizing it within chromatic-bases and the $m_\lambda$-basis. It introduces a step–route–march paradigm that relativizes CSFs across related graphs, enabling combinatorial proofs of known results and providing a concrete interpretation of the Aliste-Prieto algorithm as a special case. It proves that a forest-based chromatic-basis is a $\mathbb{Q}$-basis for the appropriate symmetric-function space and shows that, for forests, the CSF and the $U$-polynomial carry equivalent information, reinforcing a close link between these two invariants in tree- or forest-like settings. In addition, it presents a linear, morphism-based method in the $m_\lambda$-basis using induced-subgraph counts and a full-rank $\lambda$-matrix, outlining a practical route to compute CSFs from subgraph data while highlighting limitations related to reconstruction-type questions. Overall, the work provides both combinatorial insights and computational tools that bridge CSF with related graph polynomials and morphism-count frameworks.
Abstract
This paper investigates methods for calculating the chromatic symmetric function (CSF) of a graph in chromatic-bases and the $m_λ$-basis. Our key contributions include a novel approach for calculating the CSF in chromatic-bases constructed from forests and an efficient method for determining the CSF in the $m_λ$-basis. As applications, we present combinatorial proofs for two known theorems that were originally established using algebraic techniques. Additionally, we demonstrate that an algorithm introduced by Aliste-Prieto, de Mier, Orellana, and Zamora can be viewed as a case of our proposed method.