European Journal of Combinatorics ( IF 1 ) Pub Date : 2020-09-02 , DOI: 10.1016/j.ejc.2020.103205 Lech Duraj , Jakub Kozik , Dmitry Shabanov
In 1964 Erdős proved that edges are sufficient to build a -graph which is not two colorable. To this day, it is not known whether there exist such -graphs with smaller number of edges. Erdős’ bound is consequence of the fact that a hypergraph with vertices and randomly chosen edges of size is asymptotically almost surely not two colorable. Our first main result implies that for any , any -graph with randomly and uniformly chosen edges is a.a.s. two colorable. The presented proof is an adaptation of the second moment method analogous to the developments of Achlioptas and Moore (2002) who considered the problem with fixed size of edges and number of vertices tending to infinity. In the second part of the paper we consider the problem of algorithmic coloring of random -graphs. We show that quite simple, and somewhat greedy procedure, a.a.s. finds a proper two coloring for random -graphs on vertices, with at most edges. That is of the same asymptotic order as the analogue of the algorithmic barrier defined by Achlioptas and Coja-Oghlan (2008), for the case of fixed .
中文翻译:
随机超图和性质B
1964年,Erdős证明了 边缘足以建立一个 -图不是两个可着色的。迄今为止,尚不知道是否存在这样的情况。-边数较少的图形。Erdős的界线是由于以下事实的结果: 顶点和 大小随机选择的边 渐近几乎肯定不是两个可着色的。我们的第一个主要结果意味着对于任何, 任何 与图 随机且均匀选择的边缘是两个可着色的。提出的证明是第二矩方法的改编,类似于Achlioptas和Moore(2002)的发展,后者考虑了边的固定大小和趋于无穷大的顶点数量的问题。在本文的第二部分中,我们考虑了随机算法着色的问题-图。我们展示了一个非常简单且有点贪婪的过程,aas为随机找到了适当的两种颜色上的图 顶点,最多 边缘。对于固定情况,这与Achlioptas和Coja-Oghlan(2008)定义的算法障碍类似物具有相同的渐近阶。。