A non-planar version of Tutte's wheels theorem

Thtte's Wheels Theorem states that a minimally 3-connected non-wheel graph G with at least four vertices contains at least one edge e such that the contraction of e from G produces a graph which is both 3-connected and simple. The edge e is said to be non-essential. We show that a minimally 3-connected graph which is non-planar contains at least six non-essential edges. The wheel graphs are the fundamental building blocks of graphs [1]. Thtte's Wheels Theorem [7] characterizes the wheels as being the minimally 3-connected graphs with no non-essential edges. Hence a minimally 3-connected graph G that is not a wheel contains at least one non-essential edge. Such edges can be used as an important induction tool in the study of graph structure (Tutte [7]). Therefore, it is interesting to investigate the distributions of non-essential edges in minimally 3-connected graphs (see, for example, [6]). Our main result, Theorem 1, is related to Tutte's Wheels Theorem by replacing the condition that G is not a wheel in the Wheels Theorem by the condition that G is non-planar. The lower bound on the number of non-essential edges in a minimally 3-connected non-planar graph given in this theorem is best possible. Theorem 1 A minimally 3-connected non-planar graph contains at least 6 nonessential edges. The graph given in Figure 1 is a minimally 3-connected non-planar graph with only the 6 edges not appearing in triangles being non-essential. Australasian Journal of Combinatorics 20(1999), pp.3-12