Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPowers of Hamilton cycles in pseudorandom graphs
187
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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
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 $hcp(G_{n,p})={(m_1,m_2,ldots,m_r)in [0,n]^r:m_1+m_2+cdots+m_r=n}$.
We prove that if $G$ is a $k$-partite graph on $n$ vertices in which all of the parts have order at most $n/r$ and every vertex is adjacent to at least a $1-1/r+o(1)$ proportion of the vertices in every other part, then $G$ contains the $(r-1)$-st power of a Hamiltonian cycle
We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $Gamma$ the generalized Turan density $pi_F(Gamma)$ denotes the density of a maximum subgraph of $Gamma$, which contains no copy of~$F$. Extending classical Turan type results for odd cycles, we show that $pi_{F}(Gamma)=1/2$ provided $F$ is an odd cycle and $Gamma$ is a sufficiently pseudorandom graph. In particular, for $(n,d,lambda)$-graphs $Gamma$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-lambda,lambda]$, our result holds for odd cycles of length $ell$, provided [ lambda^{ell-2}ll frac{d^{ell-1}}nlog(n)^{-(ell-2)(ell-3)},. ] Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabo, and Vu, who addressed the case when $F$ is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free $(n,d,lambda)$-graphs) shows that our assumption on $Gamma$ is best possible up to the polylog-factor for every odd $ellgeq 5$.
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.
Given an $n$-vertex graph $G$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $G cup mathbb{G}(n,p)$ over the supergraphs of $G$ is referred to as a (random) {sl perturbation} of $G$. We consider the distribution of edge-coloured graphs arising from assigning each edge of the random perturbation $G cup mathbb{G}(n,p)$ a colour, chosen independently and uniformly at random from a set of colours of size $r := r(n)$. We prove that such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) geq C/n$, for some fixed $C > 0$, and $r = (1 + o(1))n$. The number of colours used is clearly asymptotically best possible. In particular, this improves upon a recent result of Anastos and Frieze (2019) in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-coloured sparse pseudo-random graphs a.a.s. admit an almost spanning rainbow path.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
