THE NUMBER OF CONTRACTIBLE EDGES IN A 4-CONNECTED GRAPH HAVING A SMALL NUMBER OF EDGES NOT CONTAINED IN TRIANGLES

Let G be a 4-connected graph, and let (E) over tilde (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 7 <= vertical bar(E) over tilde (G)vertical bar <= 8 or vertical bar(E) over tilde (G)vertical bar >= 10, then vertical bar E-c(G)vertical bar >= (vertical bar(E) over tilde (G)vertical bar + 8)/4.