Large Wk- or K3, t-Minors in 3-Connected Graphs

There are numerous results bounding the circumference of certain 3connected graphs. There is no good bound on the size of the largest bond (cocircuit) of a 3-connected graph, however. Oporowski, Oxley and Thomas [11] proved the following result in 1993. For every positive integer k, there is an integer n = f(k) such that every 3-connected graph with at least n vertices contains a Wkor K3,kminor. This result implies that the size of the largest bond in a 3-connected graph grows with the size of the graph. Oporowski et al. did not give a specific function f(k). In this paper, we first improve the above authors’ result and obtain a specific function f(k). Then we use the result to obtain a lower bound for the largest bond of a 3-connected graph. In particular, we show the following: Let G be a 3-connected planar or cubic graph on n vertices. Then for any ǫ > 0, (i) G has a Wk-minor with k = O((log n) ); and (ii) G has a bond of size at least O((log n)).