Summary
Let G bea triangle-free graph of order n and minimum degree δ > n/3. We will determine all lengths of cycles occurring in G. In particular, the length of a longest cycle or path in G is exactly the value admitted by the independence number of G. This value can be computed in time O(n 2.5) using the matching algorithm of Micali and Vazirani. An easy consequence is the observation that triangle-free non-bipartite graphs with \(\delta \geqslant \frac{3} {8}n\) are hamiltonian.
Supported by Deutsche Forschungsgemeinschaft, grant We 1265.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. Alon, Tough Ramsey graphs without short cycles, Manuscript (1993).
D. Amar, A condition for a hamiltonian graph to be bipancyclic, Discr. Math. 102(1991), 221–227.
D. Amar, I. Fournier and A. Germa, Pancyclism in Chvátal—Erdős graphs, Graphs and Combinatorics 7 (1991), 101–112.
D. Amar and Y. Manoussakis, Cycles and paths of many lengths in bipartite digraphs, J. Combin. Theory Ser. B 50 (1990), 254–264.
D. Bauer, J. van den Heuvel and E. Schmeichel, Toughness and triangle-free graphs, Preprint (1993).
C. Berge, Sur le couplage maximum d’un graphe, C. R. Acad. Sci. Paris (A) 247 (1958), 258–259.
S. Brandt, Sufficient conditions for graphs to contain all subgraphs of a given type, Ph.D. thesis, Freie Universität Berlin, 1994.
S. Brandt, R. J. Faudree and W. Goddard, Weakly pancyclic graphs, in preparation.
G. Chartrand AND L. Lesniak, Graphs & Digraphs (2nd edition), Wadsworth & Brooks/Cole, Pacific Grove, 1986.
V. Chvátal, Tough graphs and hamiltonian cycles, Discr. Math. 5 (1973), 215–228.
V. Chvátal and P. Erdős, A note on hamiltonian circuits, Discr. Math. 2 (1972), 111–113.
G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (3) 2 (1952), 69–81.
H. Enomoto, J. van den Heuvel, A. Kaneko, and A. Saito, Relative length of long paths and cycles in graphs with large degree sums, Preprint (1993).
R. Häggkvist, On the structure of non-hamiltonian graphs, Combin. Prob, and Comp. 1 (1992), 27–34.
Dingjun Lou, The Chvatal-Erdős condition for cycles in triangle-free graphs, to appear.
S. Micali and V. V. Vazirani, An O(V 1/2 E) algorithm for finding maximum matching in general graphs, Proc. 21st Ann. Symp. on Foundations of Computer Sc. IEEE, New York (1980), 17–27.
O. Ore, Note on Hamiltonian circuits, Amer. Math. Monthly 67 (1960), 55.
W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947), 107–111.
H. J. Veldman, Personal communication, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Brandt, S. (1997). Cycles and Paths in Triangle-Free Graphs. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-60406-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64393-4
Online ISBN: 978-3-642-60406-5
eBook Packages: Springer Book Archive