  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 prove that for every fixed integer $k\ge 1$, there are only
finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and
$(P_5,\overline{P_3+P_2})$-free graphs. To prove the results we use a known
structure theorem for ($P_5$,gem)-free graphs combined with properties of
$k$-vertex-critical graphs.
  Moreover, we characterize all $k$-vertex-critical ($P_5$,gem)-free graphs and
$(P_5,\overline{P_3+P_2})$-free graphs for $k \in \{4,5\}$ using a computer
generation algorithm.
