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arXiv · 2607.21568

Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs

Xichao Shu
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Abstract

We study tight Hamilton cycles in linearly quasirandom $3$-graphs. An $n$-vertex $3$-graph $H$ is $(p,\mu)$-dense if $e_H(X,Y,Z)\ge p|X||Y||Z|-\mu n^3$ for all $X,Y,Z\subseteq V(H)$. Ara\'ujo, Piga and Schacht asked whether the conditions $p,\alpha>1/4$ and $\delta_2(H)\ge\alpha n$ force a tight Hamilton cycle. We give a negative answer to this question. More generally, we determine the asymptotically sharp minimum-codegree threshold for the existence of a tight Hamilton cycle for every density $p\in(0,1)$. The resulting threshold is a discontinuous piecewise-defined function with four distinct regimes, and matching constructions show that every piece is best possible. The proof combines the absorption method and a fixed-length connecting lemma above density $1/3$ with a canonical-component Hamilton framework at and below density $1/3$.

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Xichao Shu. 2026-07-23. Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs. https://arxiv.org/abs/2607.21568

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