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Let G 3−out denote the random graph on vertex set [n] in which each vertex chooses 3 neighbors uniformly at random. Note that G 3−out has minimum degree 3 and average degree 6. We prove that the probability that G 3−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 such that the size of the largest component in G = ([n], E) is o(n) (i.e. G does not contain a giant component). Furthermore, the selection process must… (More)

Let c be a constant and (e1, f1), (e2, f2),. .. , (ecn, fcn) be a sequence of ordered pairs of edges on vertex set [n] chosen uniformly and independently at random. Let A be an algorithm for the on-line choice of one edge from each presented pair, and for i = 1,. .. , cn let GA(i) be the graph on vertex set [n] consisting of the first i edges chosen by A.… (More)

Let c be a constant and a sequence of ordered pairs of edges from the complete graph K n chosen uniformly and independently at random. We prove that there exists a constant c 2 such that if c > c 2 then whp every graph which contains at least one edge from each ordered pair (e i , f i) has a component of size Ω(n) and if c < c 2 then whp there is a graph… (More)

- Research Showcase, Cmu, Tom Bohman, Peter Keevash
- 2015

The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which… (More)

For x real, let {x} be the fractional part of x (i.e. {x} = x − −x). In this paper we prove the k = 5 case of the following conjecture (the lonely runner conjecture): for any k positive reals v 1 ,. .. , v k there exists a real number t such that 1/(k + 1) ≤ {v i t} ≤ k/(k + 1) for i = 1,. .. , k.

Let 3 ≤ k < n/2. We prove the analogue of the Erd˝ os-Ko-Rado theorem for the random k-uniform hypergraph G k (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 G k (n, p) is the size of a maximum trivial family. The analogue of the Erd˝ os-Ko-Rado theorem does not hold… (More)

Let r be a fixed constant and let H be an r-uniform, D-regular hypergraph on N vertices. Assume further that D > N ǫ for some ǫ > 0. Consider the random greedy algorithm for forming an independent set in H. An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly… (More)

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χ g (G) is the minimum k for which the first player has a winning strategy. In this… (More)