Summary
Let G be a 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 \(\delta \geq \frac{3} {8}n\) are hamiltonian.
Supported by Deutsche Forschungsgemeinschaft, grant We 1265.
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Acknowledgements
Part of this research was performed while the author was visiting the Charles University in Prague. The author would like to thank Jarík Nešetřil and his group for their kind hospitality.
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Brandt, S. (2013). Cycles and Paths in Triangle-Free Graphs. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős II. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7254-4_7
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