有向图中哈密顿圈平方的精确最小总度数阈值
The exact total degree threshold for the square of a Hamilton cycle in digraphs
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中文总结 AI 辅助
本文解决了德比亚西奥等人关于有向图中哈密顿圈平方最小总度数阈值的猜想，证明了每个足够大的\(n\)顶点有向图，当最小总度数至少为\(8n/5 - c\)（\(n\)取不同值时\(c\)取值不同）时，包含哈密顿圈的平方。
中文摘要 AI 辅助
波萨 - 西摩猜想确定了保证图中存在哈密顿圈的\(k\)次幂所需的最小度数阈值。在众多部分结果之后，科姆洛什、萨尔科齐和塞梅雷迪证实该猜想对所有足够大的图成立。特雷格洛恩后来猜测了在有向图中迫使哈密顿圈的\(k\)次幂出现的类似最小半度数阈值。随后，德比亚西奥等人针对同一问题提出了关于最小总度数阈值的猜想。本文解决了德比亚西奥等人对于\(k = 2\)的猜想。具体而言，我们证明每个具有至少\(8n/5 - c\)最小总度数的足够大的\(n\)顶点有向图包含哈密顿圈的平方，其中若\(n\equiv2,4\pmod 5\)，\(c = 2\)，否则\(c = 1\)。
英文摘要
The Pósa-Seymour conjecture establishes the minimum degree threshold required to guarantee the presence of the $k$th power of a Hamilton cycle in a graph. Following numerous partial results, Komlós, Sárközy, and Szemerédi confirmed the conjecture holds for all sufficiently large graphs. Treglown later conjectured the analogous minimum semi-degree threshold for forcing the $k$th power of a Hamilton cycle in a digraph. Subsequently, DeBiasio et al. proposed a conjecture on the minimum total degree threshold for the same problem. In this paper we settle the conjecture of DeBiasio et al. for $k=2$. Specifically, we prove that every sufficiently large $n$-vertex digraph with minimum total degree at least $8n/5-c$ contains the square of a Hamilton cycle, where $c=2$ if $n\equiv2,4\pmod 5$, and $c=1$ otherwise.
