Large W k - Or K 3, t -Minors in 3-Connected Graphs

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There are numerous results bounding the circumference of certain 3-connected graphs. There is no good bound on the size of the largest bond (cocircuit) of a 3-connected graph, however. Oporowski, Oxley, and Thomas (J Combin Theory Ser B 57 (1993), 2, 239-257) 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 Wk- or K3,k-minor. This result implies that the size of the largest bond in a 3-connected graph grows with the order of the graph. Oporowski et al. obtained a huge function f(k) iteratively. In this article, we first improve the above authors' result and provide a significantly smaller and simpler function f(k). We then use the result to obtain a lower bound for the largest bond of a 3-connected graph by showing that any 3-connected graph on n vertices has a bond of size at least 217logn. In addition, we show the following: Let G be a 3-connected planar or cubic graph on n vertices. Then for any ϵ>0, G has a Wk-minor with k=Ω((logn)1-ϵ), and thus a bond of size at least Ω((logn)1-ϵ).

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Journal of Graph Theory

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