#### Title

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

#### Document Type

Article

#### Publication Date

6-1-2016

#### Abstract

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).

#### Publication Source (Journal or Book title)

Journal of Graph Theory

#### First Page

207

#### Last Page

217

#### Recommended Citation

Ding, G., Dziobiak, S., & Wu, H.
(2016). Large W k - Or K 3, t -Minors in 3-Connected Graphs.* Journal of Graph Theory**, 82* (2), 207-217.
https://doi.org/10.1002/jgt.21895