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Consider the following stochastic graph process. We begin with G0, the empty graph on n vertices, and form Gi by adding a randomly chosen edge ei to Gi−1 where ei is chosen uniformly at random from the collection of pairs of vertices that neither appear as edges in Gi−1 nor form triangles when added as edges to Gi−1. Let the random variable M be the number(More)
Let G3−out denote the random graph on vertex set [n] in which each vertex chooses 3 neighbors uniformly at random. Note that G3−out has minimum degree 3 and average degree 6. We prove that the probability that G3−out is Hamiltonian goes to 1 as n tends to infinity.
Let e1, e2, . . . be a sequence of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). Our goal is to choose, for m as large as possible, a subset E ⊆ {e1, e2, . . . , e2m}, |E| = m, such that the size of the largest component in G = ([n], E) is o(n) (i.e. G does not contain a giant component).(More)
The H-free process, for some fixed graph H , is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we(More)
Let 3 ≤ k < n/2. We prove the analogue of the Erdős-Ko-Rado theorem for the random k-uniform hypergraph Gk(n, p) when k < (n/2)1/3; that is, we show that with probability tending to 1 as n → ∞, the maximum size of an intersecting subfamily of Gk(n, p) is the size of a maximum trivial family. The analogue of the Erdős-Ko-Rado theorem does not hold for all p(More)
Let c be a constant and (e1, f1), (e2, f2), . . . , (ecn, fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2 then whp every graph which contains at least one edge from each ordered pair (ei, fi) has a component of size Ω(n) and if(More)
Let ∆ > 1 be a fixed positive integer. For z ∈ R∆+ let Gz be chosen uniformly at random from the collection of graphs on ‖z‖1n vertices that have zin vertices of degree i for i = 1, . . . ,∆. We determine the likely evolution in continuous time of the SIR model for the spread of an infectious disease on Gz, starting from a single infected node. Either the(More)