AbstractIn this paper we compute the orientable genus of the line graph of a graph G, when G is a tree and a 2-edge connected graph, all the vertices of which have their degrees equal to 2, 3, 6, or 11 modulo 12, and either G can be imbedded with triangular faces only or G is a bipartite graph which can be imbedded with squares only as faces. In the other cases, we give an upper bound of the genus of line graphs. In this way, we solve the question of the Hamiltonian genus of the complete graph Kn, for every n ≥ 3