Cliques, minors and apex graphs

In this paper, we proved the following result: Let G be a (k + 2)-connected, non-(k − 3)apex graph where k ≥ 2. If G contains three k-cliques, say L1, L2, L3, such that |Li ∩ Lj| ≤ k − 2(1 ≤ i < j ≤ 3), then G contains a Kk+2 as a minor. Note that a graph G is t-apex if G− X is planar for some subset X ⊆ V (G) of order at most t . This theorem generalizes some earlier results by Robertson, Seymour and Thomas [N. Robertson, P.D. Seymour, R. Thomas, Hadwiger conjecture for K6-free graphs, Combinatorica 13 (1993) 279–361.], Kawarabayashi and Toft [K. Kawarabayashi, B. Toft, Any 7-chromatic graph has K7 or K4,4 as a minor, Combinatorica 25 (2005) 327–353] and Kawarabayashi, Luo, Niu and Zhang [K. Kawarabayashi, R. Luo, J. Niu, C.-Q. Zhang, On structure of k-connected graphs without Kk-minor, Europ. J. Combinatorics 26 (2005) 293–308]. © 2009 Elsevier B.V. All rights reserved.