# Constructing Hamilton cycles and perfect matchings efficiently

Anastos M. 2023. Constructing Hamilton cycles and perfect matchings efficiently. Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications. EUROCOMB: European Conference on Combinatorics, Graph Theory and Applications, 36–41.

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Abstract

<jats:p>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.</jats:p>

Publishing Year

Date Published

2023-09-01

Proceedings Title

Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications

Acknowledgement

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413.

Page

36-41

Conference

EUROCOMB: European Conference on Combinatorics, Graph Theory and Applications

Conference Location

Prague, Czech Republic

Conference Date

2023-08-28 – 2023-09-01

eISSN

IST-REx-ID

### Cite this

Anastos M. Constructing Hamilton cycles and perfect matchings efficiently. In:

*Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications*. Masaryk University Press; 2023:36-41. doi:10.5817/cz.muni.eurocomb23-005Anastos, M. (2023). Constructing Hamilton cycles and perfect matchings efficiently. In

*Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications*(pp. 36–41). Prague, Czech Republic: Masaryk University Press. https://doi.org/10.5817/cz.muni.eurocomb23-005Anastos, Michael. “Constructing Hamilton Cycles and Perfect Matchings Efficiently.” In

*Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications*, 36–41. Masaryk University Press, 2023. https://doi.org/10.5817/cz.muni.eurocomb23-005.M. Anastos, “Constructing Hamilton cycles and perfect matchings efficiently,” in

*Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications*, Prague, Czech Republic, 2023, pp. 36–41.Anastos, Michael. “Constructing Hamilton Cycles and Perfect Matchings Efficiently.”

*Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications*, Masaryk University Press, 2023, pp. 36–41, doi:10.5817/cz.muni.eurocomb23-005.**All files available under the following license(s):**

**Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0):**

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arXiv 2209.09860