  It is an open question whether the chromatic number of $t$-perfect graphs is
bounded by a constant. The largest known value for this parameter is 4, and the
only example of a 4-critical $t$-perfect graph, due to Laurent and Seymour, is
the complement of the line graph of the prism $\Pi$ (a graph is 4-critical if
it has chromatic number 4 and all its proper induced subgraphs are
3-colorable).
  In this paper, we show a new example of a 4-critical $t$-perfect graph: the
complement of the line graph of the 5-wheel $W_5$. Furthermore, we prove that
these two examples are in fact the only 4-critical $t$-perfect graphs in the
class of complements of line graphs. As a byproduct, an analogous and more
general result is obtained for $h$-perfect graphs in this class.
  The class of $P_6$-free graphs is a proper superclass of complements of line
graphs and appears as a natural candidate to further investigate the chromatic
number of $t$-perfect graphs. We observe that a result of Randerath,
Schiermeyer and Tewes implies that every $t$-perfect $P_6$-free graph is
4-colorable. Finally, we use results of Chudnovsky et al to show that
$\overline{L(W_5)}$ and $\overline{L(\Pi)}$ are also the only 4-critical
$t$-perfect $P_6$-free graphs.
