  A graph is $k$-critical if it is $k$-chromatic but each of its proper induced
subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical
$P_5$-free graphs is finite, but there is an infinite number of $k$-critical
$P_5$-free graphs for each $k \geq 5$. We show that the number of $k$-critical
$(P_5, \overline{P}_5)$-free graphs is finite for every fixed $k$. Our result
implies the existence of a certifying algorithm for $k$-coloring $(P_5,
\overline{P}_5)$-free graphs.
