### 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 |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 18 |

Issue number | 1 |

Publication status | Published - 2011 |

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### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

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

*Electronic Journal of Combinatorics*,

*18*(1).

**Toughness of K _{a,t}-minor-free graphs.** / Chen, Guantao; Egawa, Yoshimi; Kawarabayashi, Ken ichi; Mohar, Bojan; Ota, Katsuhiro.

Research output: Contribution to journal › Article

_{a,t}-minor-free graphs',

*Electronic Journal of Combinatorics*, vol. 18, no. 1.

_{a,t}-minor-free graphs. Electronic Journal of Combinatorics. 2011;18(1).

}

TY - JOUR

T1 - Toughness of Ka,t-minor-free graphs

AU - Chen, Guantao

AU - Egawa, Yoshimi

AU - Kawarabayashi, Ken ichi

AU - Mohar, Bojan

AU - Ota, Katsuhiro

PY - 2011

Y1 - 2011

N2 - 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 K3,3-minor-free graphs, we consider a generalization to a-connected Ka,t-minor-free graphs, where 3 ≤ a ≤t. We prove that there exists a positive constant h(a, t) such that every a-connected Ka,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(logh(a,t) n).

AB - 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 K3,3-minor-free graphs, we consider a generalization to a-connected Ka,t-minor-free graphs, where 3 ≤ a ≤t. We prove that there exists a positive constant h(a, t) such that every a-connected Ka,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(logh(a,t) n).

UR - http://www.scopus.com/inward/record.url?scp=79961239274&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79961239274&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79961239274

VL - 18

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -