关于严格超临界随机图的MAXCUT及随机图和随机竞赛图的着色
On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments
摘要：
我们使用一个定理丁，Lubetzky和佩雷斯描述的结构的巨组件的随机图在严格的超临界状态，以确定的典型尺寸的MAXCUT的$G\sim G\left(n,\frac {1+\varepsilon}n\right)$的$\varepsilon$。然后我们应用这个结果证明了Frieze和Pegden的猜想。对于每个$\varepsilon>0$，存在$\ell_\varepsilon$使得\whp $G\sim G(n,\frac {1+\varepsilon}n)$不同态于$2\ell_\varepsilon+1$顶点上的圈。我们还考虑了有偏随机竞赛图的着色性质。$p$ -随机竞赛图的$n$顶点是从传递竞赛图通过反转每个边独立概率$p$。我们证明了对于$p=Θ(\frac 1n)$，$p$ -随机竞赛图的色数的行为类似于具有相同边概率的随机图的色数。为了处理病例$p=\frac {1+\varepsilon}n$，我们在MAXCUT上使用上述结果。
英文摘要：
We use a theorem by Ding, Lubetzky and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of $G\sim G\left(n,\frac {1+\varepsilon}n\right)$ in terms of $\varepsilon$. We then apply this result to prove the following conjecture by Frieze and Pegden. For every $\varepsilon>0$ there exists $\ell_\varepsilon$ such that \whp $G\sim G(n,\frac {1+\varepsilon}n)$ is not homomorphic to the cycle on $2\ell_\varepsilon+1$ vertices. We also consider the coloring properties of biased random tournaments. A $p$-random tournament on $n$ vertices is obtained from the transitive tournament by reversing each edge independently with probability $p$. We show that for $p=Θ(\frac 1n)$ the chromatic number of a $p$-random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case $p=\frac {1+\varepsilon}n$ we use the aforementioned result on MAXCUT.
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