# HITTING TIME of EDGE DISJOINT HAMILTON CYCLES in RANDOM SUBGRAPH PROCESSES on DENSE BASE GRAPHS

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## Abstract

Consider the random subgraph process on a base graph G on n vertices: a sequence \{ Gt\} |tE=0(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 G0, 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 \bigl(12 + \varepsilon \bigr) 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 C\cdot n\cdot log log n and \lambda \leq c\cdot d2 for some absolute constants c, C > 0; then for a positive integer constant k with high probability log n n 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 Johansson and by Frieze and Krivelevich and answer a question posed by Frieze.

Original language English 728-754 27 SIAM Journal on Discrete Mathematics 36 1 https://doi.org/10.1137/20M1375838 Published - 2022

## Keywords

• Hamilton cycle
• graph process
• random graph

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