Better 3-coloring algorithms: excluding a triangle and a seven vertex path

We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a ${P7,C3}$-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 ${Pt,C3}$-free graphs if and only if $t \leq 7$. This implies the existence of a polynomial time certifying algorithm for list 3-coloring in ${P7,C3}$-free graphs. We furthermore determine other cases of $t, \ell$, and $k$ such that the family of minimal obstructions to list $k$-coloring in ${Pt,C{\ell}}$-free graphs is finite.