Pancyclic subgraphs of random graphs

An n‐vertex graph is called pancyclic if it contains a cycle of length t for all 3≤t≤n. In this article, we study pancyclicity of random graphs in the context of resilience, and prove that if p>n^?1/2, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than edges is pancyclic. This result is best possible in two ways. First, the range of p is asymptotically tight; second, the proportion of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich et al. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers). © 2011 Wiley Periodicals, Inc.     

Packing and counting arbitrary Hamilton cycles in random digraphs

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 ? o(1))np Hamilton cycles C₁,…,C_t, each of which is oriented arbitrarily, a digraph ～(n, p) w.h.p. contains edge disjoint copies of C₁,…,C_t, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ～(n, p) w.h.p. contains (1 ± o(1))n!pⁿ copies of C, provided .     

Compatible Hamilton cycles in random graphs

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability , the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph , an incompatibility system over G is a family where for every , the set F_v is a set of unordered pairs . An incompatibility system is Δ‐bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in F_v containing e. We say that a cycle C in G is compatible with if every pair of incident edges of C satisfies . This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant such that the random graph with is asymptotically almost surely such that for any μnp‐bounded incompatibility system over G, there is a Hamilton cycle in G compatible with . We also prove that for larger edge probabilities , the parameter μ can be taken to be any constant smaller than . These results imply in particular that typically in G(n, p) for , for any edge‐coloring in which each color appears at most μnp times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge‐coloring of such a random graph in which each color appears on at most μnp edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 533–557, 2016     

Long paths and cycles in random subgraphs of graphs with large minimum degree

For a given finite graph G of minimum degree at least k, let G_p 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 G_p asymptotically almost surely contains a cycle (and thus a path) of length at least , and (ii) if , then G_p 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     

Hamilton cycles in random graphs and directed graphs

We prove that almost every digraph D_{2–in, 2–out} is Hamiltonian. As a corollary we obtain also that almost every graph G_4–out is Hamiltonian. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 369–401, 2000     

Rainbow hamilton cycles in random graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erd?s‐Rényi random graph G_n,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 G_n,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     

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

We show that for every there exists C > 0 such that if then asymptotically almost surely the random graph contains the k^th power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi‐polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi‐polynomial algorithm for finding a tight Hamilton cycle in the random k‐uniform hypergraph for . The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of p. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.     

Large cycles in random generalized Johnson graphs

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.     

Edge‐disjoint Hamilton cycles in random graphs

We show that provided we can with high probability find a collection of edge‐disjoint Hamilton cycles in , plus an additional edge‐disjoint matching of size if is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 397–445, 2015     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

Finding cheapest cycles in vertex-weighted quasi-transitive and extended semicomplete digraphs

We consider the problem of finding a minimum cost cycle in a digraph with real-valued costs on the vertices. This problem generalizes the problem of finding a longest cycle and hence is NP-hard for general digraphs. We prove that the problem is solvable in polynomial time for extended semicomplete digraphs and for quasi-transitive digraphs, thereby generalizing a number of previous results on these classes. As a byproduct of our method we develop polynomial algorithms for the following problem: Given a quasi-transitive digraph D with real-valued vertex costs, find, for each j=1,2,…,|V(D)|, j disjoint paths P₁,P₂,…,P_j such that the total cost of these paths is minimum among all collections of j disjoint paths in D.     

Extremal subgraphs of random graphs

We prove that there is a constant c > 0, such that whenever p ≥ n^‐c, with probability tending to 1 when n goes to infinity, every maximum triangle‐free subgraph of the random graph G_n,p is bipartite. This answers a question of Babai, Simonovits and Spencer (Babai et al., J Graph Theory 14 (1990) 599–622). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with M edges, where M ? n and M ≤ /2, is “nearly unique”. More precisely, given a maximum cut C of G_n,M, we can obtain all maximum cuts by moving at most \begin{align*}\mathcal{O}(\sqrt{n^3/M})\end{align*} vertices between the parts of C. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

An algorithm for finding hamilton cycles in random directed graphs

We describe a polynomial (O(n^1.5)) time algorithm DHAM for finding hamilton cycles in digraphs. For digraphs chosen uniformly at random from the set of digraphs with vertex set {1, 2, …, n} and m = m(n) edges the limiting probability (as n → ∞) that DHAM finds a hamilton cycle equals the limiting probability that the digraph is hamiltonian. Some applications to random “travelling salesman problems” are discussed.     

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k

Let G_n,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property A_k, if G contains ⌊(k − 1)/2⌋ edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size ⌊n/2⌋. We prove that, for k ≥ 3, there is a constant C_k such that if 2m ≥ C_kn then A_k occurs in G_n,m,k with probability tending to 1 as n → ∞. © 2000 John Wiley & Sons, Inc. J. Graph Theory 34: 42–59, 2000     

Packing,counting and covering Hamilton cycles in random directed graphs

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L ^p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L ^p (R ⁿ , (φ _iρ (x))² dx) where φ _iρ is the Euclidean spherical function. The behavior is very similar to the case of the Laplace–Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.     

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?     

Counting restricted orientations of random graphs

We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.