  We present an algorithm to color a graph $G$ with no triangle and no induced
$7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is
assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this
is a special case of the problem solved in [Combinatorica 38(4):779--801,
2018], that does not require the absence of triangles, the algorithm here is
both faster and conceptually simpler. The complexity of the algorithm is
$O(|V(G)|^5(|V(G)|+|E(G)|))$, and if $G$ is bipartite, it improves to
$O(|V(G)|^2(|V(G)|+|E(G)|))$.
  Moreover, we prove that there are finitely many minimal obstructions to list
3-coloring $\{P_t,C_3\}$-free graphs if and only if $t \leq 7$. This implies
the existence of a polynomial time certifying algorithm for list 3-coloring in
$\{P_7,C_3\}$-free graphs. We furthermore determine other cases of $t, \ell$,
and $k$ such that the family of minimal obstructions to list $k$-coloring in
$\{P_t,C_{\ell}\}$-free graphs is finite.
