@article{14319, abstract = {We study multigraphs whose edge-sets are the union of three perfect matchings, M1, M2, and M3. Given such a graph G and any a1; a2; a3 2 N with a1 +a2 +a3 6 n - 2, we show there exists a matching M of G with jM \ Mij = ai for each i 2 f1; 2; 3g. The bound n - 2 in the theorem is best possible in general. We conjecture however that if G is bipartite, the same result holds with n - 2 replaced by n - 1. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.}, author = {Anastos, Michael and Fabian, David and Müyesser, Alp and Szabó, Tibor}, issn = {1077-8926}, journal = {Electronic Journal of Combinatorics}, number = {3}, publisher = {Electronic Journal of Combinatorics}, title = {{Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets}}, doi = {10.37236/11714}, volume = {30}, year = {2023}, } @inproceedings{14344, abstract = {We study the Hamilton cycle problem with input a random graph G ~ G(n,p) in two different settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one, we are given the adjacency matrix of G. In each of the two settings we derive a deterministic algorithm that w.h.p. either finds a Hamilton cycle or returns a certificate that such a cycle does not exist for p = p(n) ≥ 0. The running times of our algorithms are O(n) and respectively, each being best possible in its own setting.}, author = {Anastos, Michael}, booktitle = {Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms}, isbn = {9781611977554}, location = {Florence, Italy}, pages = {2286--2323}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Fast algorithms for solving the Hamilton cycle problem with high probability}}, doi = {10.1137/1.9781611977554.ch88}, volume = {2023}, year = {2023}, } @inproceedings{14867, 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.}, author = {Anastos, Michael}, booktitle = {Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications}, issn = {2788-3116}, location = {Prague, Czech Republic}, pages = {36--41}, publisher = {Masaryk University Press}, title = {{Constructing Hamilton cycles and perfect matchings efficiently}}, doi = {10.5817/cz.muni.eurocomb23-005}, year = {2023}, } @article{13042, abstract = {Let Lc,n denote the size of the longest cycle in G(n, c/n),c >1 constant. We show that there exists a continuous function f(c) such that Lc,n/n→f(c) a.s. for c>20, thus extending a result of Frieze and the author to smaller values of c. Thereafter, for c>20, we determine the limit of the probability that G(n, c/n)contains cycles of every length between the length of its shortest and its longest cycles as n→∞.}, author = {Anastos, Michael}, issn = {1077-8926}, journal = {Electronic Journal of Combinatorics}, number = {2}, publisher = {Electronic Journal of Combinatorics}, title = {{A note on long cycles in sparse random graphs}}, doi = {10.37236/11471}, volume = {30}, year = {2023}, } @inproceedings{12432, abstract = {We present CertifyHAM, a deterministic algorithm that takes a graph G as input and either finds a Hamilton cycle of G or outputs that such a cycle does not exist. If G ∼ G(n, p) and p ≥ 100 log n/n then the expected running time of CertifyHAM is O(n/p) which is best possible. This improves upon previous results due to Gurevich and Shelah, Thomason and Alon, and Krivelevich, who proved analogous results for p being constant, p ≥ 12n −1/3 and p ≥ 70n −1/2 respectively.}, author = {Anastos, Michael}, booktitle = {63rd Annual IEEE Symposium on Foundations of Computer Science}, isbn = {9781665455190}, issn = {0272-5428}, location = {Denver, CO, United States}, pages = {919--930}, publisher = {Institute of Electrical and Electronics Engineers}, title = {{Solving the Hamilton cycle problem fast on average}}, doi = {10.1109/FOCS54457.2022.00091}, volume = {2022-October}, year = {2022}, }