Edge-coloring problems with forbidden patterns and planted colors

Abstract: Edge-coloring problems with forbidden patterns refer to decision problems whose input is a graph $\mathbb G$ and where the goal is to determine whether the edges of $\mathbb G$ can be colored (with a fixed finite set of colors) in a way that in the resulting colored graph $(\mathbb G, \xi)$, none of a fixed set of edge-colored obstructions admits a homomorphism to $(\mathbb G, \xi)$. In the coloring extension setting, some of the edges of $\mathbb G$ are already colored and the goal is to find an extension of this coloring omitting the obstructions. We show that for certain sets of obstructions, there is a polynomial-time equivalence between the coloring problem and the extension problem. We also show that for natural sets of obstructions, such coloring problems exhibit a P vs. NP-complete complexity dichotomy.