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

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Abstract

The class of graphs with no K3,t-minors, t3, 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, t3, contains a cycle of length at least α(t)nβ. The purpose of this paper is to confirm this conjecture with α(t)=(1/2)t(t1) and β=log17292.

Keywords

Circumference
Connectivity
Graph minor
Path and cycle

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1

Supported in part by the National Security Agency of the US.

2

Supported in part by the Research Grants Council of Hong Kong, the Overseas and Hong Kong, Macau Young Scholars Collaborative Research Fund of the National Science Foundation of China, and Seed Funding for Basic Research of HKU.