The circumference of a graph with no -minor

It was shown by Chen and Yu that every 3-connected planar graph G contains a cycle of length at least | G | log 3 2 , where | G | denotes the number of vertices of G. Thomas made a conjecture in a more general setting: there exists a function β ( t ) > 0 for t ⩾ 3 , such that every 3-connected graph G with no K 3 , t -minor, t ⩾ 3 , contains a cycle of length at least | G | β ( t ) . We prove that this conjecture is true with β ( t ) = log 8 t t + 1 2 . We also show that every 2-connected graph with no K 2 , t -minor, t ⩾ 3 , contains a cycle of length at least | G | / t t − 1 .