AI 中文总结
研究n顶点4-均匀超图中哈密顿2-圈的判定问题，通过刻画满足一定条件的超图，验证猜想，给出多项式时间算法，与图的情况不同，该算法可判定特定4-均匀超图是否含哈密顿2-圈。
AI 中文摘要
在长度为t的4-均匀超图中，4-均匀2-圈是2t个顶点\(v_1v_2\cdots v_{2t}v_1\)的循环排序，使得对于\(0\leq i\leq t - 1\)（加法为模2t），\(v_{2i + 1}v_{2i + 2}v_{2i + 3}v_{2i + 4}\)是边。对于每个\(\gamma>0\)和足够大的n，我们刻画了n个顶点的4-均匀超图，其每个顶点三元组至少包含\((1/3+\gamma)n\)条边且存在哈密顿2-圈。在误差项\(\gamma n\)范围内，最小共度数假设是最优的，并验证了Garbe和Mycroft的一个猜想。这给出了一个多项式时间算法来判定最小共度数为\((1/3+\gamma)n\)的n个顶点4-均匀超图是否包含哈密顿2-圈，与图的情况形成鲜明对比。
英文摘要
A $4$-uniform $2$-cycle in a $4$-uniform hypergraph of length $t$ is a cyclic ordering of $2t$ vertices $v_1v_2\cdots v_{2t}v_1$ such that $v_{2i+1}v_{2i+2}v_{2i+3}v_{2i+4}$ are edges for $0\le i\le t-1$ while the addition is modulo $2t$. For every $γ>0$ and large $n$, we characterize the $n$-vertex $4$-uniform hypergraphs such that every triple of vertices is contained in at least $(1/3+γ)n$ edges and admits a Hamilton $2$-cycle. Up to the error term $γn$, the assumption on the minimum codegree is best possible and verifies a conjecture of Garbe and Mycroft. As a consequence, this gives a polynomial-time algorithm that decides whether an $n$-vertex $4$-uniform hypergraph with minimum codegree $(1/3+γ)n$ contains a Hamilton $2$-cycle. This stands as a steep contrast to the graph case where such a hardness gap has size $o(n)$.
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