# Long Cycles in t-Tough Graphs with t > 1

## DOI:

https://doi.org/10.51408/1963-0032## Keywords:

Hamilton cycle, Circumference, Minimum degree, Toughness## Abstract

It is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graphIt is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graph

## References

A. Bondy and U.S.R. Murty, Graph Theory with Applications, Macmillan, London and Elsevier, New York 1976.

G. A. Dirac, "Some theorems on abstract graphs", Proc. London, Math. Soc., vol. 2, pp. 69-81, 1952.

D. Bauer an d E.Schmeichel, Long Cycles in Tough Graphs, Technial Report 8612, Stevns Institute of Technology, 1986.

H.-J. Voss, // Bridges of longest circuits and of longest paths in graphs ", Beitrage zur Graphentheorie und deren Anwendungen, Vorgetr. aufdem. in t. Kolloq., Oberhof ( DDR ), pp. 275 -286, 1977.

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*Mathematical Problems of Computer Science*,

*51*, 39–56. https://doi.org/10.51408/1963-0032

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