2021
DOI: 10.48550/arxiv.2103.10414
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Tight bound for powers of Hamilton cycles in tournaments

Nemanja Draganić,
David Munhá Correia,
Benny Sudakov

Abstract: A basic result in graph theory says that any n-vertex tournament with in-and out-degrees larger than n−2 4 contains a Hamilton cycle, and this is tight. In 1990, Bollobás and Häggkvist significantly extended this by showing that for any fixed k and ε > 0, and sufficiently large n, all tournaments with degrees at least n 4 + εn contain the k-th power of a Hamilton cycle. Given this, it is natural to ask for a more accurate error term in the degree condition, and also how large n should be in the Bollobás-Häggkv… Show more

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“…An old result of Bollobás and Häggkvist [2] says that for every k and ε > 0, any tournament on sufficiently many vertices with minimum semi-degree at least (1/4 + ε)n contains a k-th power of a Hamilton cycle and this is tight up to the o(1) error term. Recently, Draganic, Munhá Correia, and Sudakov [6] were able to find an almost tight bound for the error term.…”
Section: Powers Of Cycles In ε-Intransitive Tournamentsmentioning
confidence: 99%
“…An old result of Bollobás and Häggkvist [2] says that for every k and ε > 0, any tournament on sufficiently many vertices with minimum semi-degree at least (1/4 + ε)n contains a k-th power of a Hamilton cycle and this is tight up to the o(1) error term. Recently, Draganic, Munhá Correia, and Sudakov [6] were able to find an almost tight bound for the error term.…”
Section: Powers Of Cycles In ε-Intransitive Tournamentsmentioning
confidence: 99%