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Turán numbers of theta graphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  13 February 2020

Boris Bukh  and
Michael Tait
Boris Bukh*
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA
Michael Tait
Affiliation:
Department of Mathematics and Statistics, Villanova University, Villanova, PA, 19085, USA
*
*Corresponding author. Email: bbukh@math.cmu.edu
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Abstract

The theta graph ${\Theta _{\ell ,t}}$ consists of two vertices joined by t vertex-disjoint paths, each of length $\ell $ . For fixed odd $\ell $ and large t, we show that the largest graph not containing ${\Theta _{\ell ,t}}$ has at most ${c_\ell }{t^{1 - 1/\ell }}{n^{1 + 1/\ell }}$ edges and that this is tight apart from the value of ${c_\ell }$ .

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D99: None of the above, but in this section 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 4 , July 2020 , pp. 495 - 507
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Copyright
© Cambridge University Press 2020

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Footnotes

†

Supported in part by a Sloan Research Fellowship and by US taxpayers through NSF CAREER grant DMS-1555149.

‡

This work was completed while the second author was a postdoc at Carnegie Mellon University and was supported in part by NSF grant DMS-1606350.

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