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A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In this paper we prove that every even-hole-free graph has a(More)
We consider the Erd˝ os–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3 , for some fixed λ ∈ R. We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n −1/3 , converges towards a sequence of continuous compact metric spaces. The result relies on a(More)
Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{|Mn − EMn| > x}, under quite general conditions on the branching random walk. In particular , together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89–108], our results fully(More)
We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny n. Our proof is based on a coupling which yields a precise, non-asymptotic distributional result for the case of uniformly random rooted labeled(More)
We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α log n edges where α ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question(More)
In this paper, we present new structural results about the existence of a subgraph where the degrees of the vertices are pre-specified. Further, we use these results to prove a 16-edge-weighting version of a conjecture by Karo´nski, Luczak and Thoma-son, an asymptotic 2-edge-weighting version of the same conjecture, and a 7/8 version of Louigi's Conjecture.