Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T15:26:07.946Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Number of Hamiltonian Cycles in Bipartite 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

We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 21−dd! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2)g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 4 , December 1996 , pp. 437 - 442
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]Entringer, R. C. and Swart, H. (1980) Spanning cycles of nearly cubic graphs. J. Combinatorial Theory B 29 303–309.CrossRefGoogle Scholar
[2]Jackson, B. (1993) A zero-free interval for chromatic polynomials of graphs. Combinatorics, Probability and Computing 2 325–336.CrossRefGoogle Scholar
[3]Thomason, A. G. (1978). Hamiltonian cycles and uniquely edge colourable graphs. Ann. Discrete Math. 3 259–268.CrossRefGoogle Scholar
[4]Thomassen, C. (1983) Girth in graphs. J. Combinatorial Theory B 35 129–141.CrossRefGoogle Scholar
[5]Thomassen, C. (1984) Plane representations of graphs. In: Bondy, J. A. and Murty, U. S. R. (eds), Academic Press, Toronto, 43–69.Google Scholar
[6]Tutte, W. T. (1946) On Hamiltonian circuits. J. London Math. Soc. 21 98–101.CrossRefGoogle Scholar