Cycles In Random Graphs Research Articles

Ramsey numbers of cycles in random graphs

AbstractLet be the Ramsey number of the cycle on vertices. We prove that, for some , with high probability every 2‐coloring of the edges of has a monochromatic copy of , as long as and . This is sharp up to the value of and it improves results of Letzter and of Krivelevich, Kronenberg, and Mond.

Spanning -cycles in random graphs

AbstractWe extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$ -cycles.

Color‐biased Hamilton cycles in random graphs

AbstractWe prove that a random graph , with p above the Hamiltonicity threshold, is typically such that for any r‐coloring of its edges there exists a Hamilton cycle with at least edges of the same color. This estimate is asymptotically optimal.

Hamilton cycles in random graphs with minimum degree at least 3: An improved analysis

In this paper we consider the existence of Hamilton cycles in the random graph . This random graph is chosen uniformly from , the set of graphs with vertex set [n], m edges and minimum degree at least 3. Our ultimate goal is to prove that if m = cn and c > 3/2 is constant then G is Hamiltonian w.h.p. In Frieze (2014), the second author showed that c ≥ 10 is sufficient for this and in this paper we reduce the lower bound to c > 2.662…. This new lower bound is the same lower bound found in Frieze and Pittel (2013) for the expansion of so‐called Pósa sets.

A distributed algorithm for finding Hamiltonian cycles in random graphs in [formula omitted] time

It is known for some time that a random graph G(n,p) contains w.h.p. a Hamiltonian cycle if p is larger than the critical value pcrit=(log⁡n+log⁡log⁡n+ωn)/n. The determination of a concrete Hamiltonian cycle for G(n,p) is a nontrivial task, even when p is much larger than pcrit. In this paper we consider random graphs G(n,p) with p in Ω˜(1/n), where Ω˜ hides poly-logarithmic factors in n. For this range of p we present a distributed algorithm AHC that finds w.h.p. a Hamiltonian cycle in O(log⁡n) rounds. The algorithm works in the synchronous model and uses messages of size O(log⁡n) and O(log⁡n) memory per node.

Pattern Colored Hamilton Cycles in Random Graphs

We consider the existence of patterned Hamilton cycles in randomly colored random graphs. Given a string $\Pi$ over a set of colors $\{1,2,\ldots,r\}$, we say that a Hamilton cycle is $\Pi$-colored if the pattern repeats at intervals of length $|\Pi|$ as we go around the cycle. We prove a hitting time result for the existence of such a cycle. We also prove a hitting time result for the related notion of $\Pi$-connected.

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

AbstractWe show that for every there exists C > 0 such that if then asymptotically almost surely the random graph contains the kth power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi‐polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi‐polynomial algorithm for finding a tight Hamilton cycle in the random k‐uniform hypergraph for . The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of p. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.

Geodesic cycles in random graphs

A cycle in a graph is geodesic if the distance of each pair of nodes on the cycle coincides with their distance restricted on the cycle. In this article, we prove that a random graph in G(n,p) has a geodesic cycle of length at least (2−ϵ)logdn with probability tending to one as n→∞ for d>1+ϵ with d=np and each small positive constant ϵ. This lower bound on the length of the longest geodesic cycle is almost tight since the diameter of the giant component in the random graph is asymptotically almost surely within (1±ϵ)logdn for sufficiently large d. Taking four nodes that split the geodesic cycle into four paths of approximately the same length implies that the giant component in the random graph is a.a.s. not (1∕2−ϵ)logdn-hyperbolic. This bound on the hyperbolicity improves a super-constant bound of Narayan–Saniee–Tucci and also comes close to its exact value for d≫ln5n∕(lnlnn)2 which is obtained by Mitsche and Prałat.

Combinatorial quantum gravity: geometry from random bits

I propose a quantum gravity model in which geometric space emerges from random bits in a quantum phase transition driven by the combinatorial Ollivier-Ricci curvature and corresponding to the condensation of short cycles in random graphs. This quantum critical point defines quantum gravity non-perturbatively. In the ordered geometric phase at large distances the action reduces to the standard Einstein-Hilbert term.

Finding Hamilton cycles in random graphs with few queries

AbstractWe introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of in order to typically find a subgraph possessing a given target property. We show that if , then one can find a Hamilton cycle with high probability after exposing edges. Our result is tight in both p and the number of exposed edges. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 635–668, 2016

Compatible Hamilton cycles in random graphs

AbstractA graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability , the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph , an incompatibility system over G is a family where for every , the set Fv is a set of unordered pairs . An incompatibility system is Δ‐bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in Fv containing e. We say that a cycle C in G is compatible with if every pair of incident edges of C satisfies . This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant such that the random graph with is asymptotically almost surely such that for any μnp‐bounded incompatibility system over G, there is a Hamilton cycle in G compatible with . We also prove that for larger edge probabilities , the parameter μ can be taken to be any constant smaller than . These results imply in particular that typically in G(n, p) for , for any edge‐coloring in which each color appears at most μnp times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge‐coloring of such a random graph in which each color appears on at most μnp edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 533–557, 2016

Rainbow matchings and Hamilton cycles in random graphs

AbstractLet be drawn uniformly from all m‐edge, k‐uniform, k‐partite hypergraphs where each part of the partition is a disjoint copy of . We let be an edge colored version, where we color each edge randomly from one of colors. We show that if and where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if n is even and where K is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in . Here denotes a random edge coloring of with n colors. When n is odd, our proof requires for there to be a rainbow Hamilton cycle. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 503–523, 2016

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

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.

Optimal covers with Hamilton cycles in random graphs

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o recently initiated research on the `dual' problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for log^{117}n / n < p < 1-n^{-1/8}, a.a.s. the edges of G_n,p can be covered by \lceil Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szab\'o, which holds for p > n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on packing Hamilton cycles in pseudorandom graphs.

An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

AbstractWe describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph . In this model G is drawn uniformly from graphs with vertex set [n], m edges and minimum degree at least three. We focus on the case where m = cn for constant c. If c is sufficiently large then our algorithm runs in time and succeeds w.h.p. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 73–98, 2015

Hamilton decompositions of regular expanders: Applications

In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n−1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following:(i)an undirected analogue of our result on robust outexpanders;(ii)best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree δ for a large range of values for δ.(iii)a similar result for digraphs of given minimum semidegree;(iv)an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs;(v)a verification of the ‘very dense’ case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs;(vi)a proof of a conjecture of Erdős on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.

Edge-disjoint Hamilton cycles in random graphs

We show that provided log50n/ni¾?pi¾?1-n-1/4log9n we can with high probability find a collection of i¾?i¾?G/2i¾? edge-disjoint Hamilton cycles in G~Gn,p, plus an additional edge-disjoint matching of size i¾?n/2i¾? if i¾?G is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 397-445, 2015

Rainbow hamilton cycles in random graphs

AbstractOne of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős‐Rényi random graph Gn,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs.In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of Gn,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014

On the Number of Hamilton Cycles in Sparse Random Graphs

We prove that the number of Hamilton cycles in the random graph $G(n,p)$ is $n!p^n(1+o(1))^n$ asymptotically almost surely (a.a.s.), provided that $p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}$. Furthermore, we prove the hitting time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree $2$ creates $(\frac{\ln n}{e})^n(1+o(1))^n$ Hamilton cycles a.a.s.

On a greedy 2‐matching algorithm and Hamilton cycles in random graphs with minimum degree at least three

ABSTRACTWe describe and analyse a simple greedy algorithm 2greedy that finds a good 2‐matching M in the random graph when . A 2‐matching is a spanning subgraph of maximum degree two and G is drawn uniformly from graphs with vertex set , cn edges and minimum degree at least three. By good we mean that M has components. We then use this 2‐matching to build a Hamilton cycle in time w.h.p. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 443‐497, 2014