Subdivisions of digraphs in tournaments
Abstract
We show that for every positive integer $k$, any tournament with minimum out-degree at least $(2+o(1))k^2$ contains a subdivision of the complete directed graph on $k$ vertices, which is best possible up to a factor of $8$. This may be viewed as a directed analogue of a theorem proved by Bollobás and Thomason, and independently by Komlós and Szemerédi, concerning subdivisions of cliques in graphs with sufficiently high average degree. We also consider the following problem: given $k$, what is the smallest positive integer $f(k)$ such that any $f(k)$-vertex tournament contains a $1$-subdivision of the transitive tournament on $k$ vertices? We show that $f(k)= O\left (k^2\log^3 k\right)$ which is best possible up to the logarithmic factors.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2019
- DOI:
- 10.48550/arXiv.1908.03733
- arXiv:
- arXiv:1908.03733
- Bibcode:
- 2019arXiv190803733G
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 14 pages