随机二部图的着色与依赖平面匹配
Coloured and Dependent Planar Matchings of Random Bipartite Graphs
摘要：
本文研究了随机二部图中与平面匹配有关的两个问题。首先，我们从${r}$颜色中均匀随机地对完全二部图$K_{n,n}$的每条边着色，并证明了如果${r}$与${n}$线性增长，则最大彩虹匹配是${r}$的高概率非平凡分数，即概率收敛到一，如${n \rightarrow \infty}$接下来，我们考虑平面匹配在一个依赖的设置，其中每个顶点被迫选择恰好一个邻居，从在所有可能的选择中。我们得到的最大尺寸的平面匹配的估计，并讨论了我们的结果的影响，最长的增加随机排列。
英文摘要：
In this paper, we study two problems related to planar matchings in random bipartite graphs. First, we colour each edge of the complete bipartite graph $K_{n,n}$ uniformly randomly from amongst ${r}$ colours and show that if ${r}$ grows linearly with ${n}$ then the maximum rainbow matching is a non-trivial fraction of ${r}$ with high probability, i.e. with probability converging to one as ${n \rightarrow \infty}$ Next we consider planar matchings in a dependent setting where each vertex is forced to choose exactly one neighbour from amongst all possible choices. We obtain estimates for the largest size of a planar matching and also discuss the implication of our results to longest increasing subsequences in enlarged random permutations.
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