TY - JOUR
T1 - HITTING TIME of EDGE DISJOINT HAMILTON CYCLES in RANDOM SUBGRAPH PROCESSES on DENSE BASE GRAPHS
AU - Alon, Yahav
AU - Krivelevich, Michael
N1 - Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
KW - Hamilton cycle
KW - graph process
KW - random graph
UR - http://www.scopus.com/inward/record.url?scp=85130603101&partnerID=8YFLogxK
U2 - 10.1137/20M1375838
DO - 10.1137/20M1375838
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AN - SCOPUS:85130603101
SN - 0895-4801
VL - 36
SP - 728
EP - 754
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
IS - 1
ER -