论文标题
Dirac-Type定理的横向版本的一般方法
A general approach to transversal versions of Dirac-type theorems
论文作者
论文摘要
给定超图的集合$ \ textbf {h} =（h_1，\ ldots，h_m）$，带有相同的顶点集，一个$ m $ - edge graph $ f \ subset \ subset \ cup_ {i \ in [m]} h_i $是thing h_i $，如果有一个$ ϕ：e（e（e e（e）， e（h_ {ϕ（e）}）$ in E（f）$中的每个$ e \。每个$ h_i $的最低度需要多大，以使$ \ textbf {h} $一定包含$ f $的副本，这是横向的？该集合中的每个$ h_i $都可能与超图相同，因此每个$ h_i $的最低度必须足够大，以确保$ f \ subseteq h_i $。自从Joos和Kim [Bull]的一般介绍以来。伦敦。数学。 Soc。，2020，52（3）：498-504]，越来越多的工作表明，在许多情况下，这种下限很紧。在本文中，我们通过提供广泛适用的条件来使该下限渐近地紧密地提供了统一的方法。这足以恢复该地区的许多先前结果，并获得了汉密尔顿周期的（功率）的几种经典狄拉克型结果的新型横向变体。例如，我们得出$ n $ vertex set上的任何$ rn $图的收集，每个集合至少具有至少$（r+1）+o（1））n $，包含汉密尔顿周期的$ r $ th power的横向副本。这可以看作是Pósa-Seymour猜想的彩虹版。
Given a collection of hypergraphs $\textbf{H}=(H_1,\ldots,H_m)$ with the same vertex set, an $m$-edge graph $F\subset \cup_{i\in [m]}H_i$ is a transversal if there is a bijection $ϕ:E(F)\to [m]$ such that $e\in E(H_{ϕ(e)})$ for each $e\in E(F)$. How large does the minimum degree of each $H_i$ need to be so that $\textbf{H}$ necessarily contains a copy of $F$ that is a transversal? Each $H_i$ in the collection could be the same hypergraph, hence the minimum degree of each $H_i$ needs to be large enough to ensure that $F\subseteq H_i$. Since its general introduction by Joos and Kim [Bull. Lond. Math. Soc., 2020, 52(3):498-504], a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of $rn$ graphs on an $n$-vertex set, each with minimum degree at least $(r/(r+1)+o(1))n$, contains a transversal copy of the $r$-th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa-Seymour conjecture.
