Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of Kn has w.h.p. an increasing trail of length (1-o(1))en, and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o(1).

中文翻译：

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随机图的随机边标记中的长单调轨迹

给定一张图G和一个双射F：乙(G) → {1, 2,…,e(G)}，我们说路径/路径G是F-增加如果这条路径/路径的连续边缘的标签形成一个递增序列。40 多年前，Chvátal 和 Komlós 提出了提供最坏情况估计的问题，即在ķn. Graham 和 Kleitman 解决了踪迹的案例，他们证明了答案是n-1，并且路径的情况仍然是敞开的。最近 Lavrov 和 Loh 提议研究这个问题的平均情况版本，其中边缘排序是随机均匀选择的。他们推测（Martinsson 后来证明）这样一个高概率（whp）的排序包含一条递增的 Hamilton 路径。在本文中，我们考虑随机图G=Gn,p随机均匀选择边缘排序。在此设置中，我们确定 whp 最长递增轨迹中边数的渐近线。特别是，我们证明了 Graham 和 Kleitman 结果的平均情况版本，表明ķnwhp 的长度越来越长（1-○(1))zh，而且这很紧。我们还获得了随机 Erdős-Renyi 图的最长递增路径长度的渐近紧结果p=○(1)。