The circumference of a graph with no \(K_{3,t}\)-minor (Q859611)

Let \(t\) be any integer such that \(t\geq 3\). The authors show that if \(G\) is a 3-connected graph with \(n\) nodes that contains no \(K_{3,t}\)-minors, then \(G\) has a cycle of length at least \(n^{r(t)}\), where \(r(t)= \log_{8t}(t-1)^2\); and if \(G\) is a 2-connected graph with \(n\) nodes that contains no \(K_{2,t}\)-minors, then \(G\) has a cycle of length at least \(n/t^{t-1}\).