Hitting time of edge disjoint Hamilton cycles in random subgraph processes on dense base graphs

TL;DR

This paper investigates the hitting time for the appearance of multiple edge-disjoint Hamilton cycles in random subgraphs of dense base graphs, establishing conditions under which this occurs simultaneously with minimum degree thresholds.

Contribution

It extends previous work by identifying new conditions on dense graphs that ensure the hitting time for multiple Hamilton cycles coincides with minimum degree thresholds.

Findings

Hitting time for k edge-disjoint Hamilton cycles matches minimum degree 2k in dense graphs.

Results apply to graphs with high minimum degree, bipartite expansion, and certain spectral properties.

Extends prior results and answers a question by Frieze.

Abstract

Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. There is a positive constant $\varepsilon > 0$ such that $\delta (G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. There are some constants $\alpha, \beta >0$ such that every two disjoint subsets $U,W$ of size at least $\alpha n$ have at least $\beta |U||W|$ edges between them, and the minimum degree of $G$ is at least $(2\alpha + \beta )\cdot n$; or: 3. $G$ is an $(n,d,\lambda )$--graph, with $d\geq \frac{C\cdot n\cdot \log \log n}{\log n}$ and $\lambda \leq \frac{c\cdot d^2}{n}$ for some absolute constants $c,C>0$. then for a positive integer constant $k$ with high probability the hitting time of the property of containing $k$ edge disjoint Hamilton cycles is equal to the hitting time of having minimum degree at least $2k$. These results extend prior results by by Johansson and by Frieze and Krivelevich, and answer a question posed by Frieze.