Do you want to publish a course? Click here

On the number of Hamilton cycles in sparse random graphs

184   0   0.0 ( 0 )
 Added by Roman Glebov
 Publication date 2012
  fields
and research's language is English




Ask ChatGPT about the research

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.



rate research

Read More

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C={c_1,c_2,ldots,c_r}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,ldots,m_r)in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,ldots,P_r$, where $P_i$ contains $m_i$ edges. We study $hcp(G_{n,p})$ when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when $hcp(G_{n,p})={(m_1,m_2,ldots,m_r)in [0,n]^r:m_1+m_2+cdots+m_r=n}$.
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 study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(varepsilon,p,k,ell)$-pseudorandom if for all disjoint $X$ and $Ysubset V(G)$ with $|X|gevarepsilon p^kn$ and $|Y|gevarepsilon p^ell n$ we have $e(X,Y)=(1pmvarepsilon)p|X||Y|$. We prove that for all $beta>0$ there is an $varepsilon>0$ such that an $(varepsilon,p,1,2)$-pseudorandom graph on $n$ vertices with minimum degree at least $beta pn$ contains the square of a Hamilton cycle. In particular, this implies that $(n,d,lambda)$-graphs with $lambdall d^{5/2 }n^{-3/2}$ contain the square of a Hamilton cycle, and thus a triangle factor if $n$ is a multiple of $3$. This improves on a result of Krivelevich, Sudakov and Szabo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403--426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counti
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 $ntoinfty$.
Balogh, Csaba, Jing and Pluhar recently determined the minimum degree threshold that ensures a $2$-coloured graph $G$ contains a Hamilton cycle of significant colour bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one colour). In this short note we extend this result, determining the corresponding threshold for $r$-colourings.
comments
Fetching comments Fetching comments
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا