Cycles In Random Graphs Research Articles

On the Resilience of Hamiltonicity and Optimal Packing of Hamilton Cycles in Random Graphs

Let $\bk=(k_1,...,k_n)$ be a sequence of $n$ integers. For an increasing monotone graph property $\mP$ we say that a base graph $G=([n],E)$ is \emph{$\bk$-resilient} with respect to $\mP$ if for every subgraph $H\subseteq G$ such that $d_H(i)\leq k_i$ for every $1\leq i\leq n$ the graph $G-H$ possesses $\mP$. This notion naturally extends the idea of the \emph{local resilience} of graphs recently initiated by Sudakov and Vu. In this paper we study the $\bk$-resilience of a typical graph from $\GNP$ with respect to the Hamiltonicity property where we let $p$ range over all values for which the base graph is expected to be Hamiltonian. In particular, we prove that for every $\epsilon>0$ and $p\geq\frac{\ln n+\ln\ln n +\omega(1)}{n}$ if a graph is sampled from $\GNP$ then with high probability removing from each vertex of "small" degree all incident edges but two and from any other vertex at most a $(\frac{1}{3}-\epsilon)$-fraction of the incident edges will result in a Hamiltonian graph. Considering this generalized approach to the notion of resilience allows to establish several corollaries which improve on the best known bounds of Hamiltonicity related questions. It implies that for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of $\GNP$ with respect to being Hamiltonian is at least $(1-\epsilon)np/3$, improving on the previous bound for this range of $p$. Another implication is a result on optimal packing of edge disjoint Hamilton cycles in a random graph. We prove that if $p\leq\frac{1.02\ln n}{n}$ then with high probability a graph $G$ sampled from $\GNP$ contains $\lfloor\frac{\delta(G)}{2}\rfloor$ edge disjoint Hamilton cycles, extending the previous range of $p$ for which this was known to hold.

Hamilton Cycles in Random Graphs with a Fixed Degree Sequence

Let $\mathbf{d}=d_1\leq d_2\leq\dots\leq d_n$ be a nondecreasing sequence of $n$ positive integers whose sum is even. Let $\mathcal{G}_{n,\mathbf{d}}$ denote the set of graphs with vertex set $[n]=\{1,2,\dots,n\}$ in which the degree of vertex $i$ is $d_i$. Let $G_{n,\mathbf{d}}$ be chosen uniformly at random from $\mathcal{G}_{n,\mathbf{d}}$. It will be apparent from section 4.3 that all of the sequences we are considering will be graphic. We give a condition on $\mathbf{d}$ under which we can show that whp $\mathcal{G}_{n,\mathbf{d}}$ is Hamiltonian. This condition is satisfied by graphs with exponential tails as well those with power law tails.

On two Hamilton cycle problems in random graphs

We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G n,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n.

On the Resilience of Long Cycles in Random Graphs

In this paper we determine the local and global resilience of random graphs $G_{n, p}$ ($p \gg n^{-1}$) with respect to the property of containing a cycle of length at least $(1-\alpha)n$. Roughly speaking, given $\alpha > 0$, we determine the smallest $r_g(G, \alpha)$ with the property that almost surely every subgraph of $G = G_{n, p}$ having more than $r_g(G, \alpha) |E(G)|$ edges contains a cycle of length at least $(1 - \alpha) n$ (global resilience). We also obtain, for $\alpha < 1/2$, the smallest $r_l(G, \alpha)$ such that any $H \subseteq G$ having $\deg_H(v)$ larger than $r_l(G, \alpha) \deg_G(v)$ for all $v \in V(G)$ contains a cycle of length at least $(1 - \alpha) n$ (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs.

Hamilton cycles in random graphs and directed graphs

Hamilton cycles in random graphs and directed graphs

The Random Adversary: A Lower-Bound Technique for Randomized Parallel Algorithms

The random-adversary technique is a general method for proving lower bounds on randomized parallel algorithms. The bounds apply to the number of communication steps, and they apply regardless of the processors' instruction sets, the lengths of messages, etc. This paper introduces the random-adversary technique and shows how it can be used to obtain lower bounds on randomized parallel algorithms for load balancing, compaction, padded sorting, and finding Hamiltonian cycles in random graphs. Using the random-adversary technique, we obtain the first lower bounds for randomized parallel algorithms which are provably faster than their deterministic counterparts (specifically, for load balancing and related problems).

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

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 ${\cal AR}_k$.

Cycles in random graphs and hypergraphs

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

1‐Pancyclic Hamilton Cycles in Random Graphs

AbstractWe consider a generalization of the pancyclic property. A graph G is defined to be 1‐pancyclic if there is some Hamilton cycle H in G such that we can find a cycle Cs of length s (3 ⩽ s ⩽ n − 1) using only the edges of H and one other edge es. We show that the threshold for Gn,p to be Hamiltonian, is the threshold for the 1‐pancyclic property.

Cycles in random graphs

Let G( n, p) be a graph on n vertices in which each possible edge is presented independently with probability p= p( n) and v 1( n, p) denote the number of vertices of degree 1 in G( n, p). It is shown that if ε > 0 and np( n)→∞ then the probability that G( n, p) contains cycles of all lengths r, 3⩽ r⩽ n−(1+ ε) v 1( n, p), tends to 1 as n→∞.

Pancyclic Hamilton cycles in random graphs

Let G ( n,p) denote the probability space of the set G of graphs G = ( V n , E) with vertex set V n = {1,2,…, n} and edges E chosen independently with probability p from E={{u,v}:u,v∈V n,u≠v} . A graph G∈ G ( n,p is defined to be pancyclic if, for all s, 3⩽ s⩽ n there is a cycle of size s on the edges of G. We show that the threshold probability p = ( log n + log log n + c n )/ n for the property that G contains a Hamilton cycle is also the threshold probability for the existence of a 2-pancyclic Hamilton cycle, which is defined as follows. Given a Hamilton cycle H, we will say that H is k-pancyclic if for each s (3⩽ s⩽ n−1) we can find a cycle C of length s using only the edges of H and at most k other edges.

An algorithm for finding hamilton paths and cycles in random graphs

This paper describes a polynomial time algorithm HAM that searches for Hamilton cycles in undirected graphs. On a random graph its asymptotic probability of success is that of the existence of such a cycle. If all graphs withn vertices are considered equally likely, then using dynamic programming on failure leads to an algorithm with polynomial expected time. The algorithm HAM is also used to solve the symmetric bottleneck travelling salesman problem with probability tending to 1, asn tends to ∞.

Parallel algorithms for finding Hamilton cycles in random graphs

Parallel algorithms for finding Hamilton cycles in random graphs