Edge-coloring problems with forbidden patterns are decision problems asking to find an edge-coloring of the input graph which avoids a homomorphism from a fixed forbidden family of edge-colored graphs. In the precolored version of these problems, some of the edges of the input graph are already colored, and the goal is to find an extension of this coloring which omits a homomorphism from a forbidden graph. The existence of a complexity classification for such problems is an open question of Bienvenu, ten Cate, Lutz, and Wolter (ACM TODS'14) and we answer it for certain forbidden families consisting of odd cycles and cliques. The proof consists of two main stages. First, we combine the techniques from infinite constraint satisfaction and finite Ramsey theory in order to show that the edge-coloring problem is poly-time equivalent to its precolored version. After that, we show that the precolored version is poly-time equivalent to a finite constraint satisfaction problem, which has a P vs.\ NP-complete dichotomy by the seminal results of Bulatov (FOCS'17) and Zhuk (FOCS'17).

翻译：具有禁止模式的边染色问题是判定性问题，要求为输入图找到一种边染色方案，避免来自固定禁止边染色图族的同态。在此类问题的预着色版本中，输入图的某些边已被预先着色，目标是找到该着色的扩展，并避免来自禁止图的同态。这类问题的复杂性分类存在性是由Bienvenu、ten Cate、Lutz和Wolter（ACM TODS'14）提出的一个开放问题，我们针对由奇环和团构成的特定禁止图族给出了解答。证明包含两个主要阶段。首先，我们结合无限约束满足技术与有限拉姆齐理论，证明边染色问题与其预着色版本在多项式时间内等价。之后，我们证明预着色版本与有限约束满足问题在多项式时间内等价，而后者根据Bulatov（FOCS'17）和Zhuk（FOCS'17）的开创性成果具有P vs. NP完备二分性。