Constructing Hamilton cycles and perfect matchings efficiently

EUROCOMB’23

Michael Anastos

https://doi.org/10.5817/CZ.MUNI.EUROCOMB23-005

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Abstract
Starting with the empty graph on $[n]$, at each round, a set of $K=K(n)$ edges is presented chosen uniformly at random from the ones that have not been presented yet. We are then asked to choose at most one of the presented edges and add it to the current graph. Our goal is to construct a Hamiltonian graph with $(1+o(1))n$ edges within as few rounds as possible. We show that in this process, one can build a Hamiltonian graph of size $(1+o(1))n$ in $(1+o(1))(1+(\log n)/2K) n$ rounds w.h.p. The case $K=1$ implies that w.h.p. one can build a Hamiltonian graph by choosing $(1+o(1))n$ edges in an online fashion as they appear along the first $(0.5+o(1))n\log n$ rounds of the random graph process. This answers a question of Frieze, Krivelevich and Michaeli. Observe that the number of rounds is asymptotically optimal as the first $0.5n\log n$ edges do not span a Hamilton cycle w.h.p. The case $K=\Theta(\log n)$ implies that the Hamiltonicity threshold of the corresponding Achlioptas process is at most $(1+o(1))(1+(\log n)/2K) n$. This matches the $(1-o(1))(1+(\log n)/2K) n$ lower bound due to Krivelevich, Lubetzky and Sudakov and resolves the problem of determining the Hamiltonicity threshold of the Achlioptas process with $K=\Theta(\log n)$. We also show that in the above process one can construct a graph $G$ that spans a matching of size $\lfloor V(G)/2) \rfloor$ and $(0.5+o(1))n$ edges within $(1+o(1))(0.5+(\log n)/2K) n$ rounds w.h.p. Our proof relies on a robust Hamiltonicity property of the strong $4$-core of the binomial random graph which we use as a black-box. This property allows it to absorb paths covering vertices outside the strong $4$-core into a cycle.

Pages:
36–41
References

Miklós Ajtai, János Komlós, and Endre Szemerédi. First occurrence of Hamilton cycles in random graphs. North-Holland Mathematics Studies, 115(C):173-178, 1985.
https://doi.org/10.1016/S0304-0208(08)73007-X

Michael Anastos. A note on long cycles in sparse random graphs. The Electronic Journal of Combinatorics 30.2, 2023.
https://doi.org/10.37236/11471

Michael Anastos and Alan Frieze. A scaling limit for the length of the longest cycle in a sparse random graph. Journal of Combinatorial Theory, Series B, 148:184-208, 2021.
https://doi.org/10.1016/j.jctb.2021.01.001

Tom Bohman and Alan Frieze. Avoiding a giant component. Random Structures & Algorithms, 19(1):75-85, 2001.
https://doi.org/10.1002/rsa.1019

Tom Bohman and Alan Frieze. Hamilton cycles in 3-out. Random Structures & Algorithms, 35(4):393-417, 2009.
https://doi.org/10.1002/rsa.20272

Béla Bollobás. The evolution of sparse graphs, graph theory and combinatorics, 1984. MR, 777163(2):35-57, 1984.

Alan Frieze. Hamilton cycles in random graphs: a bibliography. arXiv preprint arXiv:1901.07139, 2019.

Alan Frieze and Michal Karoński. Introduction to random graphs. Cambridge Univer- sity Press, 2016.
https://doi.org/10.1017/CBO9781316339831

Alan Frieze, Michael Krivelevich, and Peleg Michaeli. Fast construction on a restricted budget. arXiv preprint arXiv:2207.07251, 2022.

János Komlós and Endre Szemerédi. Limit distribution for the existence of Hamilto- nian cycles in a random graph. Discrete mathematics, 43(1):55-63, 1983.
https://doi.org/10.1016/0012-365X(83)90021-3

Aleksei Dmitrievich Korshunov. Solution of a problem of Erdős and Rényi on Hamil- tonian cycles in nonoriented graphs. In Doklady Akademii Nauk, volume 228, pages 529-532. Russian Academy of Sciences, 1976.

Michael Krivelevich, Eyal Lubetzky, and Benny Sudakov. Hamiltonicity thresholds in achlioptas processes. Random Structures & Algorithms, 37(1):1-24, 2010.
https://doi.org/10.1002/rsa.20302

Lajos Pósa. Hamiltonian circuits in random graphs. Discrete Mathematics, 14(4):359- 364, 1976.
https://doi.org/10.1016/0012-365X(76)90068-6

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