Tropical lines on smooth tropical surfaces

Given a tropical line L on a tropical surface X, we define its combinatorial position on X to be a certain decorated graph, showing the relative positions of vertices of X on L, and how the vertices of L are positioned on X. We classify all possible combinatorial positions of a tropical line on general smooth tropical surfaces of any degree. This classification allows one to give an upper bound for the number of tropical lines on a general smooth tropical surface of degree ≥ 3 with a given subdivision. As a concrete example, we offer a subdivision for which the associated tropical surfaces are smooth cubics with exactly 27 tropical lines in the general case, and always at least 27 tropical lines. We also give examples of smooth tropical surfaces of arbitrary degree > 3 containing no tropical lines.