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

  • Authors:

  • Guantao Chen;Laura Sheppardson;Xingxing Yu;Wenan Zang

  • Affiliations:

  • Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA and Faculty of Mathematics and Statistics, Huazhong Normal University, Wuhan, China;Department of Mathematics, University of Mississippi, MS 38677, USA;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA and Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, China;Department of Mathematics, University of Hong Kong, Hong Kong, China

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2006

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Abstract

It was shown by Chen and Yu that every 3-connected planar graph G contains a cycle of length at least |G|^l^o^g^"^3^2, where |G| denotes the number of vertices of G. Thomas made a conjecture in a more general setting: there exists a function @b(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|^@b^(^t^). We prove that this conjecture is true with @b(t)=log"8"t"^"t"^"+"^"12. 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.