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arXiv · 2311.03185

Spanning trees in pseudorandom graphs via sorting networks

Joseph Hyde·Natasha Morrison·Alp Müyesser·Matías Pavez-Signé
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Abstract

We show that $(n,d,λ)$-graphs with $λ=O(d/\log^3 n)$ are universal with respect to all bounded degree spanning trees. This significantly improves upon the previous best bound due to Han and Yang of the form $λ=d/\exp{(O(\sqrt{\log n}))}$, and makes progress towards a problem of Alon, Krivelevich, and Sudakov from 2007. Our proof relies on the existence of sorting networks of logarithmic depth, as given by a celebrated construction of Ajtai, Komlós and Szemerédi. Using this construction, we show that the classical vertex-disjoint paths problem can be solved for a set of vertices fixed in advance.

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Joseph Hyde, Natasha Morrison, Alp Müyesser, Matías Pavez-Signé. 2023-11-06. Spanning trees in pseudorandom graphs via sorting networks. https://arxiv.org/abs/2311.03185

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