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Register a new userMajority dynamics on sparse random graphs
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Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, ODonnel, Tamuz and Tan conjectured that, in the ErdH{o}s--Renyi random graph $G(n,p)$, the random initial $pm 1$-assignment converges to a $99%$-agreement with high probability whenever $p=omega(1/n)$. This conjecture was first confirmed for $pgeqlambda n^{-1/2}$ for a large constant $lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< lambda n^{-1/2}$. We break this $Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $lambda n^{-3/5}log n leq p leq lambda n^{-1/2}$ with a large constant $lambda>0$.
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We prove that the number of Hamilton cycles in the random graph G(n,p) is n!p^n(1+o(1))^n a.a.s., provided that pgeq (ln n+ln ln n+omega(1))/n. Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates (ln n/e)^n(1+o(1))^n Hamilton cycles a.a.s.
Let $Q_{n,d}$ denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at random from the subset of vectors in ${0,1}^n$ having precisely $d$ entries equal to $1$. We present a short proof of the fact that $Pr[det(Q_{n,d})=0] = Oleft(frac{n^{1/2}log^{3/2} n}{d}right)=o(1)$, whenever $d=omega(n^{1/2}log^{3/2} n)$. In particular, our proof accommodates sparse random combinatorial matrices in the sense that $d = o(n)$ is allowed. We also consider the singularity of deterministic integer matrices $A$ randomly perturbed by a sparse combinatorial matrix. In particular, we prove that $Pr[det(A+Q_{n,d})=0]=Oleft(frac{n^{1/2}log^{3/2} n}{d}right)$, again, whenever $d=omega(n^{1/2}log^{3/2} n)$ and $A$ has the property that $(1,-d)$ is not an eigenpair of $A$.
We show that for $dge d_0(epsilon)$, with high probability, the random graph $G(n,d/n)$ contains an induced path of length $(3/2-epsilon)frac{n}{d}log d$. This improves a result obtained independently by Luczak and Suen in the early 90s, and answers a question of Fernandez de la Vega. Along the way, we generalize a recent result of Cooley, Draganic, Kang and Sudakov who studied the analogous problem for induced matchings.
Resolving a conjecture of Furedi from 1988, we prove that with high probability, the random graph $G(n,1/2)$ admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which $n-o(n)$ vertices have at least as many neighbours in their own part as across. The engine of our proof is a new method to study stochastic processes driven by degree information in random graphs; this involves combining enumeration techniques with an abstract second moment argument.
We show that for any $d=d(n)$ with $d_0(epsilon) le d =o(n)$, with high probability, the size of a largest induced cycle in the random graph $G(n,d/n)$ is $(2pm epsilon)frac{n}{d}log d$. This settles a long-standing open problem in random graph theory.
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