Subgraphs with restricted degrees of their vertices in planar 3-connected graphs (Q1376060)

Every 3-connected planar graph \(G\) either contains a path with \(k\) vertices each of degree at most \(5k\) or does not contain any path with \(k\) vertices; the bound \(5k\) is the best possible. Moreover, for every connected planar graph \(H\) other than a path and for every integer \(m\geq 3\) there is a 3-connected planar graph \(G\) such that each copy of \(H\) in \(G\) contains a vertex of degree at least \(m\).