  We provide a formula for every Schur coefficient in the chromatic symmetric
function of a graph in terms of special rim hook tabloids. This formula is
useful in confirming the non-Schur positivity of the chromatic symmetric
function of a graph, especially when Stanley's stable partition method does not
work. As applications, we determine Schur positive fan graphs and Schur
positive complete tripartite graphs. We show that any squid graph obtained by
adding $n$ leaves to a common vertex on an $m$-vertex cycle is not Schur
positive if $m\ne 2n-1$, and conjecture that neither are the squid graphs with
$m=2n-1$.
