Long monotone trails in random edge-labelings of random graphs

TL;DR

This paper investigates the length of the longest increasing trails and paths in random graphs with randomly ordered edges, establishing asymptotic results and confirming conjectures about typical case behavior.

Contribution

It provides the first asymptotic analysis of the longest increasing trail and path lengths in random graphs with random edge orderings, extending classical worst-case results.

Findings

Longest increasing trail in $K_n$ whp has length $(1-o(1))en$

Longest increasing path in $G(n,p)$ with $p=o(1)$ is asymptotically tight

Confirms the average-case conjecture for Hamilton paths in random edge orderings

Abstract

Given a graph $G$ and a bijection $f : E(G)\rightarrow \{1, 2, \ldots,e(G)\}$, we say that a trail/path in $G$ is $f$-\emph{increasing} if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chv\'atal and Koml\'os raised the question of providing the worst-case estimates of the length of the longest increasing trail/path over all edge orderings of $K_n$. 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 widely open. Recently Lavrov and Loh proposed to study the average case of this problem in which the edge ordering is chosen uniformly at random. They conjectured (and it was proved by Martinsson) that such an ordering with high probability (whp) contains an increasing Hamilton path. In this paper we consider random graph $G=G(n,p)$ and its edge ordering chosen uniformly at random. In this setting we determine whp the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average case of the result of Graham and Kleitman, showing that the random edge ordering of $K_n$ has whp an increasing trail of length $(1-o(1))en$ and this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erd\H{o}-Renyi graphs with $p=o(1)$.