Distributions of Sparse Spanning Subgraphs in Random Graphs

We study the problem of finding a copy of a specific induced subgraph on inhomogeneous random graphs with infinite variance power-law degrees. We provide a fast algorithm that finds a copy of any connected graph $H$ on a fixed number of $k$ vertices as an induced subgraph in a random graph with $n$ vertices. By exploiting the scale-free graph structure, the algorithm runs in $O(n k)$ time for small values of $k$. As a corollary, this shows that the induced subgraph isomorphism problem can be solved in time $O(nk)$ for the inhomogeneous random graph. We test our algorithm on several real-world data sets.

Dense subgraphs in random graphs

For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if for a function that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and thus a path) of length at least , and (ii) if , then Gp asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015

Let H be an edge-colored hypergraph. We say that H contains a rainbow copy of a hypergraph S if it contains an isomorphic copy of S with all edges of distinct colors. We consider the following setting. A randomly edge-colored random hypergraph \(H \sim \mathcal{H}_{c}^{k}(n,p)\) is obtained by adding each k-subset of [n] with probability p, and assigning it a color from [c] uniformly, independently at random. As a first result we show that a typical \(H \sim \mathcal{H}_{c}^{2}(n,p)\) (that is, a random edge-colored graph) contains a rainbow Hamilton cycle, provided that \(c = (1 + o(1))n\) and \(p = \frac{\log n+\log \log n+\omega (1)} {n}\). This is asymptotically best possible with respect to both parameters, and improves a result of Frieze and Loh. Secondly, based on an ingenious coupling idea of McDiarmid, we provide a general tool for tackling problems related to finding “nicely edge-colored” structures in random graphs/hypergraphs. We illustrate the generality of this statement by presenting two interesting applications. In one application we show that a typical \(H \sim \mathcal{H}_{c}^{k}(n,p)\) contains a rainbow copy of a hypergraph S, provided that \(c = (1 + o(1))\vert E(S)\vert\) and p is (up to a multiplicative constant) a threshold function for the property of containment of a copy of S. In the second application we show that a typical \(G \sim \mathcal{H}_{c}^{2}(n,p)\) contains \((1 - o(1))np/2\) edge-disjoint Hamilton cycles, each of which is rainbow, provided that c = ω(n) and \(p =\omega (\log n/n)\).

We present fast and efficient randomized distributed algorithms to find Hamiltonian cycles in random graphs. In particular, we present a randomized distributed algorithm for the G(n, p) random graph model, with number of nodes n and p = c ln n/n^δ (for any constant 0 0), that finds a Hamiltonian cycle with high probability in O(n^δ) rounds. Our algorithm works in the (synchronous) CONGEST model (i.e., only O(log n)-sized messages are communicated per edge per round) and its computational cost per node is sublinear (in n) per round and is fully-distributed (each node uses only o(n) memory and all nodes' computations are essentially balanced). Our algorithm improves over the previous best known result in terms of both the running time as well as the edge sparsity of the graphs where it can succeed; in particular, the denser the random graph, the smaller is the running time.

On Matchings and Hamiltonian Cycles in Random Graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős‐Rényi random graph Gn,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs.In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of Gn,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014

We prove that the number of Hamilton cycles in the random graph $G(n,p)$ is $n!p^n(1+o(1))^n$ asymptotically almost surely (a.a.s.), provided that $p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}$. 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 $(\frac{\ln n}{e})^n(1+o(1))^n$ Hamilton cycles a.a.s.

ABSTRACTWe discuss the existence of Hamilton cycles in the random graph where there are restrictions caused by (i) coloring sequences, (ii) a subset of vertices must occur in a specific order, and (iii) there is a bound on the number of inversions in the associated permutation.

We consider the standard random geometric graph process in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of edge-length. For fixed k⩾1, weprove that the first edge in the process that creates a k-connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge-disjoint Hamilton cycles (for even k), or (k−1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge-disjoint (for odd k). This proves and extends a conjecture of Krivelevich and M **image**ler. In the special case when k = 2, our result says that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2-connectivity, which answers a question of Penrose. (This result appeared in three independent preprints, one of which was a precursor to this article.) We prove our results with lengths measured using the lp norm for any p>1, and we also extend our result to higher dimensions. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:299-322, 2011 © 2011 Wiley Periodicals, Inc.

The Small Subgraph Conditioning Method has been used to study the almost sure existence and the asymptotic distribution of the number of regular spanning subgraphs of various types in random d-regular graphs. In this paper, we use the method to determine the asymptotic distribution of the number of 3-star factors in random d-regular graphs for d ≥ 4.

Let be the random directed graph on n vertices where each of the possible arcs is present independently with probability p. A celebrated result of Frieze shows that if then typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in is typically . We also prove a hitting‐time version of this statement, showing that in the random directed graph process, as soon as every vertex has in‐/out‐degrees at least 1, there are typically directed Hamilton cycles.

Let (C1,C1′), (C2,C2′),..., (Cm,Cm′) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4(n/2 clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on-line, without backtracking, but may depend on the clauses chosen previously.We show that there exists an online choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)1/4. This contrasts with the well-known fact that a random m-clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as "Achlioptas processes." This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on-line setting above, we also consider an off-line version in which all m clause-pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off-line setting, we show that the two-choice satisfiability threshold for k-SAT for any fixed k coincides with the standard satisfiability threshold for random 2k-SAT.

On packing Hamilton cycles in [formula omitted]-regular graphs

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely ${\mathcal A \mathcal C}(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and Prałat made a major step toward this conjecture by showing that asymptotically almost surely ${\mathcal A \mathcal C}(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs.