On light graphs in the family of 4-connected planar graphs

Abstract

Let G be the family of all c-connected ( c=4 or 5) polyhedral supergraphs G of a given connected planar graph H where the mimimum vertex degree of G is 5. Let R(H) denote the maximum face size of H. We have proved for all non-empty families G : In the case R(H)<c, every G∈ G has a subgraph isomorphic to H whose vertices have a degree in G which is restricted by a number q=q(H, G) . In the case R(H)⩾c, such a restriction does not exist if H has a vertex of degree ⩾5 or if H is 3-connected.