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

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. Keywords: 4-connected graph, Contractible edge, Triangl