Abstract
There is a natural, if imprecise, notion that the requirement of trianglefreeness on a graph G forces it into a bipartite-like form. In an extreme case, if G has n vertices and edges then Turán’s Theorem gives that it must be the bipartite graph Our concern here is what happens when e is smaller, must G still exhibit bipartite-like behavior. Very roughly, our answer is: Yes, if e ≫ n 3/2, No otherwise.
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P. Erdős, Graph Theory and Probability II, Canad. J. Math 13 (1961), 346–352.
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© 1996 Springer-Verlag Berlin Heidelberg
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Erdős, P., Janson, S., Łuczak, T., Spencer, J. (1996). A Note on Triangle-Free Graphs. In: Aldous, D., Pemantle, R. (eds) Random Discrete Structures. The IMA Volumes in Mathematics and its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0719-1_8
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DOI: https://doi.org/10.1007/978-1-4612-0719-1_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6881-9
Online ISBN: 978-1-4612-0719-1
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