detailed structural theorem where the $3$-coloring properties of all the pieces are known. This is the main aim of this series of papers.

Let us first focus on the last subcase of a graph embedded in the cylinder with boundary cycles of length $4$. Note that if such a graph contains only a bounded number of separating $4$-cycles, then we can further cut the surface along them and by using the ideas of [@trfree6], we can subdivide the pieces to subpieces satisfying the conditions of one of the first two well-understood cases of the structure theorem. Hence, it is interesting to study the graphs in cylinder with many separating $4$-cycles, and this is the topic of this paper.

Let us now give a few definitions enabling us to state the main result more precisely. The *disk* is the sphere with a hole. In this paper, we generally consider graphs embedded in the sphere, the disk, or the cylinder. Furthermore, we always assume that each face of the embedding contains at most one hole, and if a face contains a hole, then the face is bounded by a cycle, which we call a *ring*. Note that the rings do not have 