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We study Hamiltonicity in random subgraphs of the hypercube $mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $kinmathbb{N}$. Secondly, we obtain a perturbation result: if $Hsubseteqmathcal{Q}^n$ satisfies $delta(H)geqalpha n$ with $alpha>0$ fixed and we consider a random binomial subgraph $mathcal{Q}^n_p$ of $mathcal{Q}^n$ with $pin(0,1]$ fixed, then with high probability $Hcupmathcal{Q}^n_p$ contains $k$ edge-disjoint Hamilton cycles, for any fixed $kinmathbb{N}$. In particular, both results resolve a long standing conjecture, posed e.g. by Bollobas, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. Our techniques also show that, with high probability, for all fixed $pin(0,1]$ the graph $mathcal{Q}^n_p$ contains an almost spanning cycle. Our methods involve branching processes, the Rodl nibble, and absorption.
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It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an active area of research, and is relatively well-understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs, and provide some results on the global behavior of the orientation of $D$. For the case that $H$ is an oriented four-cycle we prove: in every $H$-free oriented graph $D$, there is a pair $A,Bssq V(D)$ such that $e(A,B)ge e(D)^{2}/32|D|^{2}$ and $e(B,A)le e(A,B)/2$. We give a random construction which shows that this bound on $e(A,B)$ is best possible (up to the constant). In addition, we prove a similar result for the case $H$ is an oriented six-cycle, and a more precise result in the case $D$ is dense and $H$ is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph $D$, and provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.
In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $lambda_n$. Using a novel construction of subcomponents we study the largest component for $lambda_n=frac{1+chi_n}{n}$, where $epsilonge chi_nge n^{-{1/3}+ delta}$, $delta>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $chi_n=epsilon$, $| C_n^{(1)}|sim alpha(epsilon) frac{1+chi_n}{n} 2^n$ and for $o(1)=chi_nge n^{-{1/3}+delta}$, $| C_n^{(1)}| sim 2 chi_n frac{1+chi_n}{n} 2^n$ holds. This improves the result of cite{Bollobas:91} where constant $chi_n=chi$ is considered. In particular, in case of $lambda_n=frac{1+epsilon} {n}$, our analysis implies that a.s. a unique giant component exists.
Following a problem posed by Lovasz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that is, for groups $G=la a,b| a^2=1, b^s=1, (ab)^3=1, etc. ra$ generated by an involution $a$ and an element $b$ of order $sgeq3$ such that their product $ab$ has order 3. More precisely, it is shown that the Cayley graph $X=Cay(G,{a,b,b^{-1}})$ has a Hamilton cycle when $|G|$ (and thus $s$) is congruent to 2 modulo 4, and has a long cycle missing only two vertices (and thus necessarily a Hamilton path) when $|G|$ is congruent to 0 modulo 4.
Let $mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $ell$-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely $ell$ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When $r,sgeq 3$ are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Rucinski and Sileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in $mathcal{G}(n,r,s)$. Finally we prove that for $ell = 2,ldots, s-1$ and for $r$ growing moderately as $ntoinfty$, the probability that $mathcal{G}(n,r,s)$ has a $ell$-overlapping Hamilton cycle tends to zero.
An edge-ordering of a graph $G=(V,E)$ is a bijection $phi:Eto{1,2,...,|E|}$. Given an edge-ordering, a sequence of edges $P=e_1,e_2,...,e_k$ is an increasing path if it is a path in $G$ which satisfies $phi(e_i)<phi(e_j)$ for all $i<j$. For a graph $G$, let $f(G)$ be the largest integer $ell$ such that every edge-ordering of $G$ contains an increasing path of length $ell$. The parameter $f(G)$ was first studied for $G=K_n$ and has subsequently been studied for other families of graphs. This paper gives bounds on $f$ for the hypercube and the random graph $G(n,p)$.
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