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 Bonomo et al. (2018) [1], 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,C₃}-free graphs if and only if t≤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,ℓ, and k such that the family of minimal obstructions to list k-coloring in {P_t,C_ℓ}-free graphs is finite.