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 L 1 , L 2 , L 3 , such that | L i ∩ L j | ≤ k − 2 ( 1 ≤ i < j ≤ 3 ) , then G contains a K k + 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 . heorem generalizes some earlier results by Robertson, Seymour and Thomas [N. Robertson, P.D. Seymour, R. Thomas, Hadwiger conjecture for K 6 -free graphs, Combinatorica 13 (1993) 279–361.], Kawarabayashi and Toft [K. Kawarabayashi, B. Toft, Any 7-chromatic graph has K 7 or K 4 , 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 K k -minor, Europ. J. Combinatorics 26 (2005) 293–308].