# Non-hamiltonian Graphs with Given Toughness

## Keywords:

Hamilton cycle, Toughness of graph## Abstract

In 1973, Chvátal introduced the concept of toughness τ of a graph and conjectured that there exists a finite constant t0 such that every t0-tough graph (that is τ ≥ t0) is hamiltonian. To solve this challenging problem, all efforts are directed towards constructing non-hamiltonian graphs with toughness as large as possible. The last result in this direction is due to Bauer, Broersma and Veldman, which states that for each positive ϵ, there exists a non-hamiltonian graph with 9/4 - ϵ ≤ τ < 9/4. The following related broad-scale problem, reminding the well-known pancyclicity or hypohamiltonicity, arises naturally: whedher there exists a non-hamiltonian graph with a given toughness. We conjecture that if there exist a non-hamiltonian t-tough graph then for each rational number a with 0 < a ≤ t there exists a non-hamiltonian graph whose toughness is exactly a. In this paper we prove this conjecture for t = 9/4 - ϵ by using a number of additional modified building blocks to construct the required graphs.

## References

D. Bauer, H. J. Broersma, J. van den Heuvel and H. J. Veldman, “On Hamiltonian properties of 2-tough graphs", J. Graph Theory, vol. 18, pp. 539-543, 1994.

D. Bauer, H. J. Broersma and H. J. Veldman, “Not every 2-tough graph is hamiltonian", Discrete Appl. Math., vol. 99, pp. 317-321, 2000.

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V. Chvátal, “Tough graphs and Hamiltonian circuits", Discrete Math., vol. 5, pp. 215-228, 1973.

H. Enomoto, B. Jackson, P. Katerinis and A. Saito, “Toughness and the existence of k-factors", J. Graph Theory, vol. 9, pp. 87-95, 1985.

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*Mathematical Problems of Computer Science*,

*40*, 13–22. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/244

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