scholarly journals Random Perturbation of Sparse Graphs

10.37236/9510 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Max Hahn-Klimroth ◽  
Giulia Maesaka ◽  
Yannick Mogge ◽  
Samuel Mohr ◽  
Olaf Parczyk
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Random Perturbation ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Fixed Constant ◽  
Bounded Degree Trees ◽  
Random Part

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

scholarly journals Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

2016 ◽  
Vol 25 (6) ◽  
pp. 909-927 ◽  
Author(s):  
MICHAEL KRIVELEVICH ◽  
MATTHEW KWAN ◽  
BENNY SUDAKOV
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Random Perturbation ◽  
Hamilton Cycle ◽  
Regularity Lemma ◽  
Hamilton Cycles ◽  
Interesting Application ◽  
Discrete Structures ◽  
Szemeredi's Regularity Lemma ◽  
Strong Expansion

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a densek-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

scholarly journals Packing Hamilton Cycles Online

2018 ◽  
Vol 27 (4) ◽  
pp. 475-495
Author(s):  
JOSEPH BRIGGS ◽  
ALAN FRIEZE ◽  
MICHAEL KRIVELEVICH ◽  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Hitting Time ◽  
Online Version ◽  
Hamilton Cycles ◽  
Fixed Integer ◽  
Edge Disjoint ◽  
Colour Class

It is known that w.h.p. the hitting time τ2σ for the random graph process to have minimum degree 2σ coincides with the hitting time for σ edge-disjoint Hamilton cycles [4, 9, 13]. In this paper we prove an online version of this property. We show that, for a fixed integer σ ⩾ 2, if random edges of Kn are presented one by one then w.h.p. it is possible to colour the edges online with σ colours so that at time τ2σ each colour class is Hamiltonian.

scholarly journals Matchings and Hamilton cycles in hypergraphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
International Audience

International audience It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.

scholarly journals The power of two choices for random walks

2021 ◽  
pp. 1-28
Author(s):  
Agelos Georgakopoulos ◽  
John Haslegrave ◽  
Thomas Sauerwald ◽  
John Sylvester
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Graph ◽  
Optimal Strategy ◽  
Cover Time ◽  
Bounded Degree ◽  
Graph Classes ◽  
Algorithmic Question ◽  
Cover Times ◽  
Bounded Degree Trees

Abstract We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

scholarly journals An Undecidability Result on Limits of Sparse Graphs

10.37236/2340 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Endre Csóka
Keyword(s):  
Random Graph ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Random Node ◽  
Bounded Degree Graphs ◽  
Undecidability Result

Given a set $\mathcal{B}$ of finite rooted graphs and a radius $r$ as an input, we prove that it is undecidable to determine whether there exists a sequence $(G_i)$ of finite bounded degree graphs such that the rooted $r$-radius neighbourhood of a random node of $G_i$ is isomorphic to a rooted graph in $\mathcal{B}$ with probability tending to 1. Our proof implies a similar result for the case where the sequence $(G_i)$ is replaced by a unimodular random graph.

scholarly journals Strong Games Played on Random Graphs

10.37236/5414 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Asaf Ferber ◽  
Pascal Pfister
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Perfect Matching ◽  
Hamilton Cycle ◽  
Target Structure ◽  
Fixed Constant ◽  
Matching Game

In a strong game played on the edge set of a graph $G$ there are two players, Red and Blue, alternating turns in claiming previously unclaimed edges of $G$ (with Red playing first). The winner is the first one to claim all the edges of some target structure (such as a clique $K_k$, a perfect matching, a Hamilton cycle, etc.). In this paper we consider strong games played on the edge set of a random graph $G\sim G(n,p)$ on $n$ vertices. We prove that $G\sim G(n,p)$ is typically such that Red can win the perfect matching game played on $E(G)$, provided that $p\in(0,1)$ is a fixed constant. 

scholarly journals Random Graph's Hamiltonicity is Strongly Tied to its Minimum Degree

10.37236/8339 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Yahav Alon ◽  
Michael Krivelevich
Keyword(s):  
Random Graph ◽  
Analogous Result ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Perfect Matchings ◽  
Delta G

We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.

scholarly journals Multicoloured Hamilton cycles in random graphs; an anti-Ramsey threshold

10.37236/1213 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Hamilton Cycle ◽  
Hamilton Cycles

Let the edges of a graph $G$ be coloured so that no colour is used more than $k$ times. We refer to this as a $k$-bounded colouring. We say that a subset of the edges of $G$ is multicoloured if each edge is of a different colour. We say that the colouring is $\cal H$-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ${\cal AR}_k$ = $\{G :$ every $k$-bounded colouring of $G$ is $\cal H$-good$\}$. We establish the threshold for the random graph $G_{n,m}$ to be in ${\cal AR}_k$.

scholarly journals Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs

10.37236/5025 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Asaf Ferber
Keyword(s):  
Random Graph ◽  
Directed Graph ◽  
Random Graphs ◽  
High Probability ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Edge Probability

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for graphs on an even number of vertices, where for odd $n$ their $p$ was $\omega((\log n)/n)$. Lastly, we show that for $p=(1+o(1))(\log n)/n$, if we randomly color the edge set of a random directed graph $D_{n,p}$ with $(1+o(1))n$ colors, then w.h.p. one can find a rainbow Hamilton cycle where all the edges are directed in the same way.

scholarly journals Hamilton Cycles in Sparse Robustly Expanding Digraphs

10.37236/7418 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Allan Lo ◽  
Viresh Patel
Keyword(s):  
Minimum Degree ◽  
Directed Graphs ◽  
Hamilton Cycle ◽  
Regularity Lemma ◽  
Hamilton Cycles ◽  
Szemeredi's Regularity Lemma

The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs. These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle. Previous proofs of this require Szemerédi's Regularity Lemma and so this fact can only be applied to dense, sufficiently large robust expanders. We give a proof that does not use the Regularity Lemma and, indeed, we can apply our result to sparser robustly expanding digraphs.

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