Structure of edges in a 4-connected graph not contained in triangles and the number of contractible edges

Abstract Let G be a 4-connected graph, and let E ( G ) denote the set of those edges of G which are not contained in a triangle, and let E c ( G ) denote the set of 4-contractible edges of G . We show that if 3 ≤ | E ( G ) | ≤ 4 or | E ( G ) | ≥ 7 , then | E c ( G ) | ≥ ( | E ( G ) | + 8 ) ∕ 4 unless G has one of the three specified configurations.