  We obtain the Schur positivity of spider graphs of the forms $S(a,2,1)$ and
$S(a,4,1)$, which are considered to have the simpliest structures for which the
Schur positivity was unknown. The proof outline has four steps. First, we find
noncommutative analogs for the chromatic symmetric functions of the spider
graphs $S(a,b,1)$. Secondly, we expand the analogs under the $\Lambda$- and
$R$-bases, whose commutative images are the elementary and skew Schur symmetric
functions, respectively. Thirdly, we recognize the Schur coefficients via the
Littlewood--Richardson rule in terms of norms of multisets of Yamanouchi words.
At last we establish the Schur positivity combinatorially together with the aid
of computer assistance.
