AI 中文总结
该论文确定多部图遍历中保证存在哈密顿遍历的临界密度随部数增大趋于1/2，解决了相关猜想，还得到了几类图的遍历F-因子子图的渐近临界密度，证明用了吸收法。
AI 中文摘要
设$G$为$r$部图，且任意两部之间的边密度至少为$\alpha$。我们研究需多大的$\alpha$才能保证$G$存在哈密顿遍历（即含每部恰好一个顶点的$r$-圈子图），并证明该临界密度随$r$增大趋于$\frac{1}{2}$，这解决了Badakhshian、Falgas-Ravry和Sharifzadeh提出的猜想。我们还研究保证遍历中存在其他生成结构（尤其是子图因子）所需的临界密度，得到了几类图$F$的遍历$F$-因子子图的渐近临界密度，结果的证明涉及吸收法。
英文摘要
Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $α$. We consider the problem of determining how large $α$ must be in order to guarantee that $G$ has a Hamiltonian traversal (an $r$-cycle subgraph containing exactly one vertex from each part), and show that this critical density tends to $\frac 1 2$ as $r$ increases. This resolves a conjecture of Badakhshian, Falgas-Ravry, and Sharifzadeh. We also study the critical densities necessary to guarantee the existence of other spanning structures in traversals, particularly subgraph factors, and obtain asymptotically the critical densities for traversal $F$-factor subgraphs for several classes of graphs $F$. The proofs of our results involve the absorption method.
Comments27 pages, 2 figures
