Light subgraphs of multigraphs embedded in compact 2-manifolds

Fabrici and Jendrolʹ proved that each 3-connected plane graph with a k-path (a path of k vertices), contains a k-path such that each vertex has a degree at most 5k. We generalized these results to manifolds M of Euler characteristic χ(M) < 0; here a 2-manifold is always a compact 2-dimensional manifold.