Four gravity results

The gravity of a graph H   in a given family of graphs HH is the greatest integer n with the property that for every integer m  , there exists a supergraph G∈HG∈H of H such that each subgraph of G, which is isomorphic to H, contains at least n   vertices of degree ⩾m⩾m in G  . Madaras and Škrekovski introduced this concept and showed that the gravity of the path PkPk on k⩾2k⩾2 vertices in the family of planar graphs of minimum degree 2 is k-2k-2 for each k≠5,7,8,9k≠5,7,8,9. They conjectured that for each of the four excluded cases the gravity is k-3k-3. In this paper we show that this holds.