arXiv subjects

Nicholas Wormald

Publications and source records attributed to Nicholas Wormald.

28 records · Page 2Linked to original sources

arXiv2010

The diameter of sparse random graphs

In this paper we study the diameter of the random graph $G(n,p)$, i.e., the the largest finite distance between two vertices, for a wide range of functions $p=p(n)$. For $p=\la/n$ with $\la>1$ constant, we give a simple proof of an essentially best possible result, with an $O_p(1)$ additive correction term. Using similar techniques, we establish 2-point concentration in the case that $np\to\infty$. For $p=(1+ε)/n$ with $ε\to 0$, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an $O_p(1/ε)$ additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph $G(n,p)$ to an accuracy of the order of its standard deviation (or better), for all functions $p=p(n)$. Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.

math.PR↗

arXiv2009

An improved upper bound on the length of the longest cycle of a supercritical random graph

We improve Luczak's upper bounds on the length of the longest cycle in the random graph G(n,M) in the "supercritical phase" where M=n/2+s and s=o(n) but n^{2/3}=o(s). The new upper bound is (6.958+o(1))s^2/n with probability 1-o(1) as n approaches infinity. Letting c=1+2s/n, the equivalence between G(n,p) and G(n,M) implies the same result for G(n,p) where p=c/n, c approaching 1, c-1 = omega(n^{-1/3}).

math.CO↗

arXiv2008

On the chromatic number of random d-regular graphs

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k-1 or k. If moreover d>(2k-3)log(k-1), then the value k-1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small subgraph conditioning method to the number of balanced k-colourings, where a colouring is balanced if the number of vertices of each colour is equal.

math.CO↗

arXiv2008

High degree graphs contain large-star factors

We show that any finite simple graph with minimum degree $d$ contains a spanning star forest in which every connected component is of size at least $Ω((d/\log d)^{1/3})$. This settles a problem of J. Kratochvil.

math.CO↗

arXiv2008

Regular induced subgraphs of a random graph

An old problem of Erdős, Fajtlowicz and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on n vertices. Motivated by this problem, we consider the order of such a subgraph in a typical graph on n vertices, i.e., in a binomial random graph G(n,1/2). We prove that with high probability a largest induced regular subgraph of G(n,1/2) has about n^{2/3} vertices.

math.CO↗

arXiv2007

Induced forests in regular graphs with large girth

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.

math.CO↗

arXiv2007

On the threshold for k-regular subgraphs of random graphs

The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a random graph is at most the threshold for the $(k+2)$-core. In particular, this pins down the point of appearance of a $k$-regular subgraph in $\G(n,p)$ to a window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$.

math.CO↗

arXiv2007

Expansion properties of a random regular graph after random vertex deletions

We investigate the following vertex percolation process. Starting with a random regular graph of constant degree, delete each vertex independently with probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away from 0. We show that a.a.s. the resulting graph has a connected component of size n-o(n) which is an expander, and all other components are trees of bounded size. Sharper results are obtained with extra conditions on alpha. These results have an application to the cost of repairing a certain peer-to-peer network after random failures of nodes.

math.CO↗

arXiv2007

On the hardness of sampling independent sets beyond the tree threshold

We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity $ł$, where the weight of an independent set $I$ is $ł^{|I|}$. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree $d$ and $ł<ł_c(d)$, where $ł_c(d)$ is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the $d$-regular infinite tree. We show that for $d \geq 3$, $ł$ just above $ł_c(d)$ with high probability over $d$-regular bipartite graphs, any local Markov chain Monte-Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for ``replica'' method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. We conjecture that $ł_c$ is in fact the exact threshold for this computational problem, i.e., that for $ł>ł_c$ it is NP-hard to approximate the above weighted sum overindependent sets to within a factor polynomial in the size of the graph.

math.PR↗

arXiv2005

Rainbow Hamilton cycles in random regular graphs

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.

math.CO↗