Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-07T23:20:58.698Z Has data issue: false hasContentIssue false
  • English
  • Français

Minors in Graphs with High Chromatic Number

Published online by Cambridge University Press:  13 April 2011

THOMAS BÖHME ,
ALEXANDR KOSTOCHKA  and
ANDREW THOMASON
THOMAS BÖHME
Affiliation:
Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany (e-mail: [email protected])
ALEXANDR KOSTOCHKA
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA and Sobolev Institute of Mathematics, Novosibirsk, Russia (e-mail: [email protected])
ANDREW THOMASON
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop lower bounds on the Hadwiger number h(G) of graphs G with high chromatic number. In particular, if G has n vertices and chromatic number k then h(G) ≥ (4k − n)/3.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 4 , July 2011 , pp. 513 - 518
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Dirac, G. A. (1964) Homomorphism theorems for graphs. Math. Ann. 153 69–80.CrossRefGoogle Scholar
[2]Duchet, P. and Meyniel, H. (1982) On Hadwiger's number and the stability number. In Graph Theory (Bollobás, B., ed.), Vol. 13 of Annals of Discrete Mathematics, North-Holland, pp. 71–73.Google Scholar
[3]Gallai, T. (1963) Kritische Graphen II. Publ. Math. Inst. Hungar. Acad. Sci. 8 373–395.Google Scholar
[4]Kawarabayashi, K. and Song, Z. (2007) Independence numbers and clique minors. J. Graph Theory 56 219–226.CrossRefGoogle Scholar
[5]Plummer, M. D., Stiebitz, M. and Toft, B. (2003) On a special case of Hadwiger's conjecture. Discussiones Mathematicae Graph Theory 23 333–363.CrossRefGoogle Scholar
[6]Robertson, N., Seymour, P. D. and Thomas, R. (1993) Hadwiger's conjecture for K 6-free graphs. Combinatorica 13 279–362.CrossRefGoogle Scholar
[7]Stehlík, M. (2009) Critical graphs with connected complements. J. Combin. Theory Ser. B 89 189–194.CrossRefGoogle Scholar
[8]Wagner, K. (1937) Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114 570–590.CrossRefGoogle Scholar