Lower Bound on the Number of Contractible Edges in a 4-Connected Graph with Edges Not Contained in Triangles

Abstract

Let G be a 4-connected graph, and let $$\tilde{E}(G)$$ 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 $$|\tilde{E}(G)| \ge 1$$ , then $$|E_{c}(G)| \ge (|\tilde{E}(G)|+8)/4$$ unless G has one of the specified configurations.