Linear relations on star coefficients of the chromatic symmetric function

TL;DR

This paper establishes a link between specific coefficients in the chromatic symmetric function and graph connectivity, introduces new linear relations among these coefficients, and explores their connections to graph orientations.

Contribution

It proves that certain star coefficients determine 2-connectedness of graphs, introduces new linear relations, and finds new bases for spans of chromatic symmetric functions.

Findings

Coefficient of star st_{21^{n-2}} determines 2-connectedness

New linear relations among star coefficients

Relation between star coefficient and acyclic orientations

Abstract

We prove that the coefficient of the star $\mathfrak{st}_{21^{n-2}}$ in the chromatic symmetric function $X_G$ determines whether a connected graph $G$ is $2$-connected. We also prove new linear relations on other star coefficients of chromatic symmetric functions. This allows us to find new bases for certain spans of chromatic symmetric functions. Finally, we relate the coefficient of the star $\mathfrak{st}_n$ to acyclic orientations.