Hamilton Cycles in Random Regular Digraphs

SIAM Journal on Discrete Mathematics, 2013

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 p ≥ ln n+ln ln n+ω 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.

The Electronic Journal of Combinatorics, 2015

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for g...

The Electronic Journal of Combinatorics, 2004

Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − 1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.

Random Structures and Algorithms, 2009

We show that there is a constant c so that for fixed r ≥ 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c √ n vertices as a minor. This confirms a conjecture of Markström . Since any minor of an r-regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph Gn,p during the phase transition (i.e. when pn → 1).

Lecture Notes in Computer Science, 2014

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε, p, k, ℓ)-pseudorandom if for all disjoint X and Y ⊆ V (G) with |X| ≥ εp k n and |Y | ≥ εp ℓ n we have e(X, Y) = (1 ± ε)p|X||Y |. We prove that for all β > 0 there is an ε > 0 such that an (ε, p, 1, 2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n, d, λ)-graphs with λ ≪ 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 Szabó [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 counting versions.

Ninth International Conference on Computer Science and Information Technologies Revised Selected Papers, 2013

Let D be a strong digraph on n ≥ 4 vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) d(x) + d(y) ≥ 2n − 1 and min{d + (x) + d − (y), d − (x) + d + (y)} ≥ n − 1 for every pair of non-adjacent vertices x, y with a common in-neighbour or a common out-neighbour, then D is hamiltonian. In this note we show that: if D is not directed cycle and satisfies the condition (*), then D contains a cycle of length n − 1 or n − 2.

Random Structures and Algorithms, 2001

Random d-regular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random d-regular graph when d = d n grows more quickly than √ n. These properties include connectivity, hamiltonicity, independent set size, chromatic number, choice number, and the size of the second eigenvalue, among others.

2001

An oriented graph is an out-tournament if the out-neighbourhood of every vertex is a tournament. This note is motivated by A. Kemnitz and B. Greger, Gongr. Numer. 130 (1998) 127-131. We show that the main result of the paper by Kemnitz and Greger is an easy consequence of the characterization of hamiltonian out-tournaments by Bang-Jensen, Huang and Prisner, J. Gombin. Theory Ser. B 59 (1993) 267-287. We also disprove a conjecture from the paper of Kemnitz and Greger.

European Journal of Combinatorics, 2006

For a graph G the random n-lift of G is obtained by replacing each of its vertices by a set of n vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of G are adjacent. We show that asymptotically almost surely the random lift of a graph G is hamiltonian, provided G has the minimum degree at least 5 and contains two disjoint Hamiltonian cycles whose union is not a bipartite graph.