Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userColorful Hamilton cycles in random graphs
237
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $hcp(G_{n,p})={(m_1,m_2,ldots,m_r)in [0,n]^r:m_1+m_2+cdots+m_r=n}$.
rate research
Read More
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.
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
Posas theorem states that any graph $G$ whose degree sequence $d_1 le ldots le d_n$ satisfies $d_i ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e. we prove a `resilience version of Posas theorem: if $pn ge C log n$ and the $i$-th vertex degree (ordered increasingly) of $G subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Diracs theorem obtained by Lee and Sudakov. Chvatals theorem generalises Posas theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvatals theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.
In an $r$-uniform hypergraph on $n$ vertices a tight Hamilton cycle consists of $n$ edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of $r$ vertices. We provide a first deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton cycles in random $r$-uniform hypergraphs with edge probability at least $C log^3n/n$. Our result partially answers a question of Dudek and Frieze [Random Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton cycles exists already for $p=omega(1/n)$ for $r=3$ and $p=(e + o(1))/n$ for $rge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Bottcher, Kohayakawa and Person [Random Structures & Algorithms 46 (2015), 446-465] and Nenadov and v{S}koric [arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a randomised polynomial time algorithm working for edge probabilities $pge n^{-1+varepsilon}$, while the algorithm of Nenadov and v{S}koric is a randomised quasipolynomial time algorithm working for edge probabilities $pge Clog^8n/n$.
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
