  In this note we obtain numerous new bases for the algebra of symmetric
functions whose generators are chromatic symmetric functions. More precisely,
if $\{ G_ k \}_{k\geq 1}$ is a set of connected graphs such that $G_k$ has $k$
vertices for each $k$, then the set of all chromatic symmetric functions $\{
X_{G_ k} \}_{k\geq 1}$ generates the algebra of symmetric functions. We also
obtain explicit expressions for the generators arising from complete graphs,
star graphs, path graphs and cycle graphs.
