Toughness of Ka,t-minor-free graphs

Guantao Chen, Yoshimi Egawa, Ken ichi Kawarabayashi, Bojan Mohar, Katsuhiro Ota

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 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).

Original languageEnglish
JournalElectronic Journal of Combinatorics
Volume18
Issue number1
Publication statusPublished - 2011

Fingerprint

Toughness
Minor
Planar graph
Connected graph
Graph in graph theory
Number of Components
Substructure
Spanning tree
Maximum Degree
Walk
Corollary
Odd
Cycle
Vertex of a graph

ASJC Scopus subject areas

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

Cite this

Chen, G., Egawa, Y., Kawarabayashi, K. I., Mohar, B., & Ota, K. (2011). Toughness of Ka,t-minor-free graphs. Electronic Journal of Combinatorics, 18(1).

Toughness of Ka,t-minor-free graphs. / Chen, Guantao; Egawa, Yoshimi; Kawarabayashi, Ken ichi; Mohar, Bojan; Ota, Katsuhiro.

In: Electronic Journal of Combinatorics, Vol. 18, No. 1, 2011.

Research output: Contribution to journalArticle

Chen, G, Egawa, Y, Kawarabayashi, KI, Mohar, B & Ota, K 2011, 'Toughness of Ka,t-minor-free graphs', Electronic Journal of Combinatorics, vol. 18, no. 1.
Chen G, Egawa Y, Kawarabayashi KI, Mohar B, Ota K. Toughness of Ka,t-minor-free graphs. Electronic Journal of Combinatorics. 2011;18(1).
Chen, Guantao ; Egawa, Yoshimi ; Kawarabayashi, Ken ichi ; Mohar, Bojan ; Ota, Katsuhiro. / Toughness of Ka,t-minor-free graphs. In: Electronic Journal of Combinatorics. 2011 ; Vol. 18, No. 1.
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