The Classi cation of Finitely Spreading Graphs

Thomassen recently introduced the concept of a nitely spreading graph: an innnite graph whose edges can be oriented, each in one, both, or neither direction, so that every vertex has nite out-degree and every ray has a forward oriented tail. He conjectured that a graph is nitely spreading if and only if it is bounded|equivalently, if it contains none of three speciied innnitely spreading graphs. We prove Thomassen'sconjecture in amended form, adding a fourth minimal obstruction to the three conjectured ones.