  This article is dedicated to the study of positivity phenomena for the
chromatic symmetric function of a graph with respect to various bases of
symmetric functions. We give a new proof of Gasharov's theorem on the
Schur-positivity of the chromatic symmetric function of a $(3 + 1)$-free poset.
We present a combinatorial interpretation of the Schur-coefficients in terms of
planar networks. Compared to Gasharov's proof, it gives a clearer visual
illustration of the cancellation procedures and is quite similar in spirit to
the proof of monomial positivity of Schur functions via the
Lindstrom-Gessel-Viennot lemma. We apply a similar device to the $e$-positivity
problem of chromatic functions. Following Stanley, we analyze certain analogs
of symmetric functions attached to graphs instead of working with chromatic
symmetric functions of graphs directly. We introduce a new combinatorial
object: the correct sequences of unit interval orders, and, using these, we
reprove monomial positivity of $G$-analogues of the power sum symmetric
functions.
