  Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it
contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the
path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic
number $k$ but every proper induced subgraph of $G$ has chromatic number less
than $k$. The study of $k$-vertex-critical graphs for graph classes is an
important topic in algorithmic graph theory because if the number of such
graphs that are in a given hereditary graph class is finite, then there is a
polynomial-time algorithm to decide if a graph in the class is
$(k-1)$-colorable.
  In this paper, we initiate a systematic study of the finiteness of
$k$-vertex-critical graphs in subclasses of $P_5$-free graphs. Our main result
is a complete classification of the finiteness of $k$-vertex-critical graphs in
the class of $(P_5,H)$-free graphs for all graphs $H$ on 4 vertices. To obtain
the complete dichotomy, we prove the finiteness for four new graphs $H$ using
various techniques -- such as Ramsey-type arguments and the dual of Dilworth's
Theorem -- that may be of independent interest.
