Rainbow Hamilton cycles in random graphs and hypergraphs

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.

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

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 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

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

We extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$ -cycles.

We describe sufficient conditions for the existence of Hamiltonian paths in oriented graphs and use these to provide a complete description of the tournaments with no two edge-disjoint Hamiltonian paths. We prove that tournaments with small irregularity have many edge-disjoint Hamiltonian cycles in support of Kelly's conjecture.

The n-dimensional crossed cube CQn, a variation of the hypercube Qn, has the same number of vertices and the same number of edges as Qn, but it has only about half of the diameter of Qn. In the interconnection network, some efficient communication algorithms can be designed based on edge-disjoint Hamiltonian cycles. In addition, two edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant Hamiltonicity for the interconnection network. Hung [Discrete Applied Mathematics 181, 109–122, 2015] designed a recursive algorithm to construct two edge-disjoint Hamiltonian cycles on CQn in O(n2n) time. In this paper, we provide an O(n) time algorithm for each vertex in CQn to determine which two edges were used in Hamiltonian cycles 1 and 2, respectively. With the information of each vertex, we can construct two edge-disjoint Hamiltonian cycles in CQn with n ≥ 4.

The n-dimensional locally twisted cube LTQn, a variation of the hypercube Qn, has the same number of vertices and the same number of edges as Qn, but it has only about half of the diameter of Qn. The existence of the Hamiltonian cycle provides an advantage when implementing algorithms that require a ring structure. In addition, k (≥ 2) edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant Hamiltonicity for the interconnection network. Hung [Theoretical Computer Science 412, 4747–4753, 2011] proved that LTQn with n ⩾ 4 contains two edge-disjoint Hamiltonian cycles, and posted an open problem what is the maximum number of edge-disjoint Hamiltonian cycles in LTQn for n ⩾ 6? In this paper, we show that there exist three edge-disjoint Hamiltonian cycles on LTQn while n ⩾ 6.

The random-adversary technique is a general method for proving lower bounds on randomized parallel algorithms. The bounds apply to the number of communication steps, and they apply regardless of the processors' instruction sets, the lengths of messages, etc. This paper introduces the random-adversary technique and shows how it can be used to obtain lower bounds on randomized parallel algorithms for load balancing, compaction, padded sorting, and finding Hamiltonian cycles in random graphs. Using the random-adversary technique, we obtain the first lower bounds for randomized parallel algorithms which are provably faster than their deterministic counterparts (specifically, for load balancing and related problems).

Recognizing random intersection graphs

Ore proved in 1960 that if G is a graph of order n and the sum of the degrees of any pair of nonadjacent vertices is at least n, then G has a hamiltonian cycle. In 1986, Li Hao and Zhu Yongjin showed that if n ⩾ 20 and the minimum degree δ is at least 5, then the graph G above contains at least two edge disjoint hamiltonian cycles. The result of this paper is that if n ⩾ 2δ2, then for any 3 ⩽ l1 ⩽ l2 ⩽ ⃛ ⩽ lk ⩽ n, 1 = k = [(δ ‐ 1)/2], such graph has K edge disjoint cycles with lengths l1, l2…lk, respectively. In particular, when l1 = l2 = ⃛ = lk = n and k = [(δ ‐ 1)/2], the graph contains [(δ ‐ 1)/2] edge disjoint hamiltonian cycles.

Hamiltonian cycles play an important role in graph theory and data mining applications. Two Hamiltonian cycles that don’t have an edge in common are known as edge-disjoint Hamiltonian cycles (EDHCs). EDHCs are useful in computer networks. They have found applications in improving network capacity, fault-tolerance and collusion resistant mining algorithms. This paper extends previous work on collusion resistance capability of data mining algorithms. We first propose a new trust model for network computers. We then use this model as a basis to improve the collusion resistance capability of data mining algorithms. We use a performance metric to quantify the improvement.

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.