Search for Hamiltonian Cycles

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.

Proceedings of the fifth annual ACM symposium on Parallel algorithms and architectures - SPAA '93, 1993

We give tight bounds on the parallel complexity of some problems involving random graphs. Speci cally, w e show that a Hamiltonian cycle, a breadth rst spanning tree, and a maximal matching can all be constructed in log n expected time using n= log n processors on the CRCW PRAM. This is a substantial improvement o ver the best previous algorithms, which required log log n 2 time and n log 2 n processors. We then introduce a technique which allows us to prove that constructing an edge cover of a random graph from its adjacency matrix requires log n expected time on a CRCW PRAM with On processors. Constructing an edge cover is implicit in constructing a spanning tree, a Hamiltonian cycle, and a maximal matching, so this lower bound holds for all these problems, showing that our algorithms are optimal. This new lower bound technique is one of the very few lower bound techniques known which apply to randomized CRCW PRAM algorithms, and it provides the rst nontrivial parallel lower bounds for these problems.

2000

We consider the problem of finding long paths and cycles in Hamiltonian graphs. The focus of our work is on sparse graphs, e.g., cubic graphs, that satisfy some property known to hold for Hamiltonian graphs, e.g., k-cyclability. We first consider the problem of finding long cycles in 3-connected cubic graphs whose edges have weights wi _> 0. We find cycles of weight at least (~ w~) ~ for a = log~ 3. Based on this result, we develop an algorithm for finding a cycle of length at least m 0°g3 2)/2 in 5-cyclable graphs with vertices of degree at most 3. We then consider 3-cyelable graphs with vertices of degree at most 3, and show how to find a cycle of length at least 2~. We consider the graph property of 1-toughness that is common to Hamiltonian graphs and 3-connected cubic graphs, and try to determine if 1toughness implies the existence of long cycles. We show that 2-connectivity and 1-toughness, for constant degree graphs, may give cycles that are only of logarithmic length. However, we exhibit a class of 3-connected 1-tough graphs with degrees up to 6 where we can find cycles of length at least ml°ga 2/2. 1 * E-maih t omas@t heory, stanford, edu.

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...

SpringerPlus, 2016

The Hamiltonian cycle (HC) problem has many applications such as time scheduling, the choice of travel routes and network topology . Therefore, resolving the HC is an important problem in graph theory and computer science as well . It is known to be in the class of NP-complete problems and consequently, determining if a graph is Hamiltonian, using the current algorithms, if it has a high time complexity. The difficulty of finding HC increases exponentially with the problem size. Also, there is an algorithm for solving the HC problem with polynomial expected running time . A Hamiltonian path is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle is the cycle that visits each vertex once. A Hamiltonian graph is a graph that has a Hamiltonian cycle . Due to their similarities, the problem of an HC is usually compared with Euler's problem, but solving them is very different. There exists a very elegant, necessary and sufficient condition for a graph to have Euler Cycles. Also, literature presents many solutions that generate efficient algorithms for finding Euler Cycles. Unfortunately, there is a lack of solutions for the Hamilton problem. In addition to the author' knowledge, there is no research indicating a necessary and sufficient condition for a graph to have a HC. Some of the theorems provide finding sufficient conditions for a graph to be Hamiltonian

Commentarii Mathematici Helvetici, 1969

Computing Research Repository, 2000

We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and random directed graphs, D. If n is finite, we assume that G or D contains a hamilton circuit. If G is an arbitrary graph containing a hamilton circuit, we conjecture that Algorithm G always obtains a hamilton circuit in polynomial time.

international symposium on algorithms and computation, 2016

We present a matching and LP based heuristic algorithm that decides graph non-Hamiltonicity. Each of the n! Hamilton cycles in a complete directed graph on n + 1 vertices corresponds with each of the n! n-permutation matrices P , such that p u,i = 1 if and only if the i th arc in a cycle enters vertex u, starting and ending at vertex n + 1. A graph instance (G) is initially coded as exclusion set E, whose members are pairs of components of P , {p u,i , p v,i+1 }, i = 1, n − 1, for each arc (u, v) not in G. For each {p u,i , p v,i+1 } ∈ E, the set of P satisfying p u,i = p v,i+1 = 1 correspond with a set of cycles not in G. Accounting for all arcs not in G, E codes precisely the set of cycles not in G. A doubly stochastic-like O(n 4) formulation of the Hamilton cycle decision problem is then constructed. Each {p u,i , p v,j } is coded as variable q u,i,v,j such that the set of integer extrema is the set of all permutations. We model G by setting each q u,i,v,j = 0 in correspondence with each {p u,i , p v,j } ∈ E such that for non-Hamiltonian G, integer solutions cannot exist. We then recognize non-Hamiltonicity by iteratively deducing additional q u,i,v,j that can be set zero and expanding E until the formulation becomes infeasible, in which case we recognize that no integer solutions exists i.e. G is decided non-Hamiltonian. The algorithm first chooses any {p u,i , p v,j } ∈ E and sets q u,i,v,j = 1. As a relaxed LP, if the formulation is infeasible, we deduce q u,i,v,j = 0 and {p u,i , p v,j } can be added to E. Then we choose another {p u,i , p v,j } ∈ E and start over. Otherwise, as a subset of matching problems together with a subset of necessary conditions, if q u,i,v,j cannot participate in a match, we deduce q u,i,v,j = 0 and {p u,i , p v,j } can be added E. We again choose another {p u,i , p v,j } ∈ E and start over. Otherwise q u,i,v,j is undecided, and we exhaustively test all {p u,i , p v,j } ∈ E. If E becomes the set of all {p u,i , p v,j }, G is decided non-Hamiltonian. Otherwise G is undecided. We call this the Weak Closure Algorithm. Only non-Hamiltonian G share this maximal property. Over 100 non-Hamiltonian graphs (10 through 104 vertices) and 2000 randomized 31 vertex non-Hamiltonian graphs are tested and correctly decided non-Hamiltonian. For Hamiltonian G, the complement of E provides information about covers of matchings, perhaps useful in searching for cycles. We also present an example where the WCA fails to deduce any integral value for any q u,i,v,j i.e. G is undecided.

Random Structures & Algorithms, 2018

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,…,Ct, each of which is oriented arbitrarily, a digraph ∼(n, p) w.h.p. contains edge disjoint copies of C1,…,Ct, 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 .

SIAM Journal on Discrete Mathematics, 2007

We show for an arbitrary ℓp norm that the property that a random geometric graph G(n, r) contains a Hamiltonian cycle exhibits a sharp threshold at r = r(n) = log n αp n , where αp is the area of the unit disk in the ℓp norm. The proof is constructive and yields a linear time algorithm for finding a Hamiltonian cycle of G(n, r) a.a.s., provided r = r(n) ≥ log n (αp−ǫ)n for some fixed ǫ > 0.