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Register a new userThe largest hole in sparse random graphs
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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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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 $lambda>0$.
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.
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 $L$ be subset of ${3,4,dots}$ and let $X_{n,M}^{(L)}$ be the number of cycles belonging to unicyclic components whose length is in $L$ in the random graph $G(n,M)$. We find the limiting distribution of $X_{n,M}^{(L)}$ in the subcritical regime $M=cn$ with $c<1/2$ and the critical regime $M=frac{n}{2}left(1+mu n^{-1/3}right)$ with $mu=O(1)$. Depending on the regime and a condition involving the series $sum_{l in L} frac{z^l}{2l}$, we obtain in the limit either a Poisson or a normal distribution as $ntoinfty$.
Let D(G) be the smallest quantifier depth of a first order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the first order descriptive complexity of G. We will show that almost surely D(G)=Theta(ln n/lnln n), where G is a random tree of order n or the giant component of a random graph G(n,c/n) with constant c>1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.
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