$(2k+1)$-connected tournaments with large minimum out-degree are $k$-linked
Abstract
Pokrovskiy conjectured that there is a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that any $2k$-strongly-connected tournament with minimum out and in-degree at least $f(k)$ is $k$-linked. In this paper, we show that any $(2k+1)$-strongly-connected tournament with minimum out-degree at least some polynomial in $k$ is $k$-linked, thus resolving the conjecture up to the additive factor of $1$ in the connectivity bound, but without the extra assumption that the minimum in-degree is large. Moreover, we show the condition on high minimum out-degree is necessary by constructing arbitrarily large tournaments that are $(2.5k-1)$-strongly-connected but are not $k$-linked.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2019
- DOI:
- 10.48550/arXiv.1912.00710
- arXiv:
- arXiv:1912.00710
- Bibcode:
- 2019arXiv191200710G
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 15 pages, 2 figures