Dirac图的鲁棒Hamilton性
Robust Hamiltonicity of Dirac graphs
摘要：
一个图是哈密尔顿的，如果它包含一个圈，它通过图的每个顶点只一次。Dirac在1952年提出的一个经典定理证明了$n$顶点上的最小度至少为$n/2$的图是Hamilton图。我们把这样的图称为狄拉克图。在本文中，我们在两个方向上推广了狄拉克定理，并证明了狄拉克图是一个非常强意义下的鲁棒Hamilton图。首先，考虑Dirac图的随机子图，证明了存在一个常数$p$使得如果$p \ge C \log n / n$，则a.a.s.所得到的随机子图仍然是哈密尔顿的。其次，我们证明了在Dirac图上进行$(1:b)$ Maker-Breaker博弈时，只要对某个绝对常数$c > 0$，偏差$b$至多为$cn /\log n$，Maker就能构造出一个Hamilton子图.这两个结果都是紧到一个常数因子，并在一个一般框架下得到证明。
英文摘要：
A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac's theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability $p$, and prove that there exists a constant $C$ such that if $p \ge C \log n / n$, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a $(1:b)$ Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias $b$ is at most $cn /\log n$ for some absolute constant $c > 0$. Both of these results are tight up to a constant factor, and are proved under one general framework.
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