arXiv subjects

Colin McDiarmid

Publications and source records attributed to Colin McDiarmid.

44 records · Page 3Linked to original sources

arXiv2011

Connectivity for bridge-addable monotone graph classes

A class A of labelled graphs is bridge-addable if for all graphs G in A and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and u is also in A; the class A is monotone if for all G in A and all subgraphs H of G, H is also in A. We show that for any bridge-addable, monotone class A whose elements have vertex set 1,...,n, the probability that a uniformly random element of A is connected is at least (1-o_n(1)) e^{-1/2}, where o_n(1) tends to zero as n tends to infinity. This establishes the special case of a conjecture of McDiarmid, Steger and Welsh when the condition of monotonicity is added. This result has also been obtained independently by Kang and Panagiotiou (2011).

math.CO↗

arXiv2011

On the chromatic number of random geometric graphs

Given independent random points $X_1,...,X_n\in\eR^d$ with common probability distribution $ν$, and a positive distance $r=r(n)>0$, we construct a random geometric graph $G_n$ with vertex set $\{1,...,n\}$ where distinct $i$ and $j$ are adjacent when $\norm{X_i-X_j}\leq r$. Here $\norm{.}$ may be any norm on $\eR^d$, and $ν$ may be any probability distribution on $\eR^d$ with a bounded density function. We consider the chromatic number $χ(G_n)$ of $G_n$ and its relation to the clique number $ω(G_n)$ as $n \to \infty$. Both McDiarmid and Penrose considered the range of $r$ when $r \ll (\frac{\ln n}{n})^{1/d}$ and the range when $r \gg (\frac{\ln n}{n})^{1/d}$, and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the `phase change' range when $r \sim (\frac{t\ln n}{n})^{1/d}$ with $t>0$ a fixed constant. Both McDiarmid and Penrose asked for the behaviour of the chromatic number in this range. We determine constants $c(t)$ such that $\frac{χ(G_n)}{nr^d}\to c(t)$ almost surely. Further, we find a "sharp threshold" (except for less interesting choices of the norm when the unit ball tiles $d$-space): there is a constant $t_0>0$ such that if $t \leq t_0$ then $\frac{χ(G_n)}{ω(G_n)}$ tends to 1 almost surely, but if $t > t_0$ then $\frac{χ(G_n)}{ω(G_n)}$ tends to a limit $>1$ almost surely.

math.CO↗

arXiv2010

Random graphs with few disjoint cycles

The classical Erdős-Pósa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erdős-Pósa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B.

math.CO↗

arXiv2010

The t-stability number of a random graph

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).

math.CO↗

arXiv2009

The t-improper chromatic number of random graphs

We consider the $t$-improper chromatic number of the Erd{\H o}s-R{é}nyi random graph $G(n,p)$. The t-improper chromatic number $χ^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $χ^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.

math.CO↗

arXiv2007

Asymptotic distributions and chaos for the supermarket model

In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate λn, where 0<λ<1. Each customer chooses d > 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n -> oo. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n^{-1}; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n^{-1}.

math.PR↗

arXiv2006

On the maximum queue length in the supermarket model

There are $n$ queues, each with a single server. Customers arrive in a Poisson process at rate $λn$, where $0<λ<1$. Upon arrival each customer selects $d\geq2$ servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as $n\to\infty$ the maximum queue length takes at most two values, which are $\ln\ln n/\ln d+O(1)$.

math.PR↗

arXiv2005

On the power of two choices: Balls and bins in continuous time

Suppose that there are n bins, and balls arrive in a Poisson process at rate λn, where λ>0 is a constant. Upon arrival, each ball chooses a fixed number d of random bins, and is placed into one with least load. Balls have independent exponential lifetimes with unit mean. We show that the system converges rapidly to its equilibrium distribution; and when d\geq 2, there is an integer-valued function m_d(n)=\ln \ln n/\ln d+O(1) such that, in the equilibrium distribution, the maximum load of a bin is concentrated on the two values m_d(n) and m_d(n)-1, with probability tending to 1, as n\to \infty. We show also that the maximum load usually does not vary by more than a constant amount from \ln \ln n/\ln d, even over quite long periods of time.

math.PR↗