### Abstract

The toughness of a non-complete graph G is the minimum value of among all separating vertex sets S ⊂ V(G), where ω(G - S) ≥ 2 is the number of components of G - S. It is well-known that every 3-connected planar graph has toughness greater than 1/2. Related to this property, every 3-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a 2-walk, etc. Realizing that 3-connected planar graphs are essentially the same as 3-connected K_{3,3}-minor-free graphs, we consider a generalization to a-connected K_{a,t}-minor-free graphs, where 3 ≤ a ≤t. We prove that there exists a positive constant h(a, t) such that every a-connected K_{a,t}-minor-free graph G has toughness at least h(a,t). For the case where a = 3 and t is odd, we obtain the best possible value for h(3, t). As a corollary it is proved that every such graph of order n contains a cycle of length Ω(log_{h(a,t)} n).

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 18 |

Issue number | 1 |

Publication status | Published - 2011 Aug 12 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

_{a,t}-minor-free graphs.

*Electronic Journal of Combinatorics*,

*18*(1).