Abstract
Pokrovskiy conjectured that there is a function f: ℕ → 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 tournaments but are not k-linked.
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The authors would like to thank the anonymous referees for their comments and suggestions which helped to clarify some of the arguments.
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The first author wishes to acknowledge support by the EPSRC, grant. no. EP/N019504/1.
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Girão, A., Popielarz, K. & Snyder, R. (2K + 1)-Connected Tournaments with Large Minimum Out-Degree are K-Linked. Combinatorica 41, 815–837 (2021). https://doi.org/10.1007/s00493-021-4374-3
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DOI: https://doi.org/10.1007/s00493-021-4374-3