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