  In 1995 Stanley introduced a generalization of the chromatic polynomial of a
graph $G$, called the chromatic symmetric function, $X_G$, which was
generalized to noncommuting variables, $Y_G$, by Gebhard-Sagan in 2001.
Recently there has been a renaissance in the study of $X_G$, in particular in
classifying when $X_G$ is a positive linear combination of elementary symmetric
or Schur functions.
  We extend this study from $X_G$ to $Y_G$, including establishing the
multiplicativity of $Y_G$, and showing $Y_G$ satisfies the $k$-deletion
property. Moreover, we completely classify when $Y_G$ is a positive linear
combination of elementary symmetric functions in noncommuting variables, and
similarly for Schur functions in noncommuting variables, in the sense of
Bergeron-Hohlweg-Rosas-Zabrocki. We further establish the natural
multiplicative generalization of the fundamental theorem of symmetric
functions, now in noncommuting variables, and obtain numerous new bases for
this algebra whose generators are chromatic symmetric functions in noncommuting
variables. Finally, we show that of all known symmetric functions in
noncommuting variables, only all elementary and specified Schur ones can be
realized as $Y_G$ for some $G$.
