On Light Graphs in 3-Connected Plane Graphs Without Triangular or Quadrangular Faces (original) (raw)

We prove that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path P k , a path on k vertices, such that each of its k vertices has degree 5=3k in G or does not contain any k-path. We also prove that each 3-connected pentagonal plane graph G which has a k-cycle, a cycle on k vertices, k P f5; 8; 11; 14g, contains a k-cycle such that all its vertices have, in G, bounded degrees. Moreover, for all integers k and m, k ! 3, k = P f5; 8; 11; 14g and m ! 3, we present a graph in which every k-cycle contains a vertex of degree at least m.