The circumference of a graph with no K3, t-minor, II

The class of graphs with no K3,t-minors, t ≥ 3, contains all planar graphs and plays an important role in graph minor theory. In 1992, Seymour and Thomas conjectured the existence of a function α(t) > 0 and a constant β > 0, such that every 3-connected n-vertex graph with no K3,t-minors, t ≥ 3, contains a cycle of length at least α(t)n . The purpose of this paper is to confirm this conjecture with α(t) = (1/2)t(t−1) and β = log1729 2. ∗Supported in part by NSA and by NSFC Project 10628102 †Supported in part by the Research Grants Council of Hong Kong and Seed Funding for Basic Research of HKU.

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