On vertex-degree restricted paths in polyhedral graphs Original Research Article

It is proved that every 3-connected planar graph G with δ(G)⩾4 either does not contain any path on k⩾8 vertices or must contain a path on k vertices (k⩾8) having degree (in G) at most 5k−7; the bound 5k−7 is shown to be the best possible. For every connected planar graph H different from a path and for every integer m⩾4 there is a 3-connected planar graph G with δ(G)⩾4 such that each subgraph of G isomorphic to H has a vertex x with degG(x)⩾m.