Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-24T10:10:56.062Z Has data issue: false hasContentIssue false
  • English
  • Français

K5-Subdivisions in Graphs

Published online by Cambridge University Press:  12 September 2008

Carsten Thomassen
Carsten Thomassen
Affiliation:
Mathematical Institute, Technical University of Denmark, DK-2800 Lyngby, Denmark
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1964 Dirac conjectured that every graph with n vertices and at least 3n − 5 edges contains a subdivision of K5 We prove a weakened version with 7/2;n − 7 instead of 3n − 5. We prove that, for any prescribed vertex νo, the subdivision can be found such that νo is not one of the five branch vertices. This variant of Dirac's problem, which was suggested by the present author in 1973, is best possible in the sense that 7/2;n − 7 cannot be replaced by 7/2;n − 15/2;.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 2 , June 1996 , pp. 179 - 189
Copyright
Copyright © Cambridge University Press 1996

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]Bondy, J. A. and Lovász, L. (1981) Cycles through specified vertices of a graph. Combinatoric 1 114–140.CrossRefGoogle Scholar
[2]Bollobás, B. (1978) Extremal Graph Theory, Academic Press.Google Scholar
[3]Bollobás, B. and Thomason, A. G.Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs, Combinatorica, to appear.Google Scholar
[4]Dirac, G. A. (1964) Homomorphism theorems for graphs, Math. Ann. 153 69–80.CrossRefGoogle Scholar
[5]Komlós, J. and Szemerédi, E.Topological cliques in graphs, Combinatorics, Probability and Computing, to appear.Google Scholar
[6]Lovász, L. (1974) Problem 5, Period. Math. Hungar. 4 82.Google Scholar
[7]Perfect, H. (1968) Applications of Menger's theorem, J. Math. Analysis Appl. 22 96–111.CrossRefGoogle Scholar
[8]Seymour, P. D. (1980) Disjoint paths in graphs, Discrete Math. 29 292–309.CrossRefGoogle Scholar
[9]Thomassen, C. (1974) Some homeomorphism properties of graphs, Math. Nachr. 64 119–133.CrossRefGoogle Scholar
[10]Thomassen, C. (1980) 2-linked graphs, Euro. J. Combinatorics 1 371–378.CrossRefGoogle Scholar
[11]Thomassen, C. (1984) Plane representations of graphs, in Progress in Graph Theory, Bondy, J. A. and Murty, U. S. R., eds., Academic Press, pp. 43–69.Google Scholar
[12]Thomassen, C. (1988) Paths circuits and subdivisions, in Selected Topics in Graph Theory, Beineke, L. W. and Wilson, R. J., eds., Academic Press, pp. 97–131.Google Scholar