Hamiltonian cycle

• I think there are some applications in electronic circuit design/construction; for example Abstract: Multi-threshold CMOS (MTCMOS) is currently the most popular methodology in industry for implementing a power gating design, which can effectively reduce the leakage power by turning off inactive circuit domains. However, large peak current may be consumed in a power-gated domain during its sleep-to-active mode transition. As a result, major IC foundries recommend turning on power switches one by one to reduce the peak current during the mode transition, which requires a Hamiltonian-cycle routing to serially connect all the power switches. ... • Another """application""" (note the triple quotes :-) is puzzle games ... for example in the game RoundTrip (a.k.a. GrandTour) you must find an Hamiltonian circuit in a grid of points in which some of the edges are given. But there are many other puzzles/videogames that are directly inspired by the Hamiltonian circuit/path problem: Inertia, Pearl, Rolling Cube Puzzles, Slither,... ... and the "hardness" of HC makes them addictive: even small instances can be very hard to solve for our brain!!! Thanks for contributing an answer to Theoretical Computer Science Stack Exchange! • Please be sure to answer the question. Provide details and share your research! But avoid … • Asking for help, clarification, or responding to other answers. • Making statements based on opinion; back them up with references or personal experience. Use MathJax...

Determining if a graph has a check if a given path is a Hamiltonian cycle in polynomial time, but we don't know any polynomial time algorithms capable of finding it. The only algorithms that can be used to find a Hamiltonian cycle are exponential time algorithms. Some of them are • Brute force search • Dynamic programming • Other exponential but nevertheless faster algorithms that you can find

This article is about the specific problem of determining whether a Hamiltonian path or cycle exists in a given graph. For the general graph theory concepts, see In the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a The Hamiltonian cycle problem is a special case of the n (if so, the route is a Hamiltonian circuit; if there is no Hamiltonian circuit then the shortest route will be longer). Reduction between the path problem and the cycle problem [ ] The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: • In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than finding a Hamiltonian cycle. • In the other direction, the Hamiltonian cycle problem for a graph G is equivalent to the Hamiltonian path problem in the graph H obtained by adding terminal ( s and t attached respectively to a vertex v of G and to v', a v which gives v' the same neighbourhood as v. The Hamiltonian path in H running through vertices s − v − x − ⋯ − y − v ′ − t . Algorithms [ ] There are n! different sequences of vertices that might be Hamiltonian paths in a given n-vertex graph (and are, in a n 22 n). In this method, one determines, for each set S of verti...

You may want to download the 1. Hamiltonian Paths and Cycles This video defines and illustrates examples of Hamiltonian paths and cycles. We explore the question of whether we can determine whether a graph has a Hamiltonian cycle, and certificates for a “yes” answer. (10:45) 2. Certificates for “No” Answer Given a graph G, there does not seem to be a way to provide a certificate to validate a “no” answer to the question: Does G have a Hamiltonian cycle? However, there are exceptions. (3:37) 3. Computational Complexity 1: P We introduce, and provide examples of, the class P that consists of all “yes-no” questions for which the answer can be determined using an algorithm which is provably correct and has a running time which is polynomial in the input size. (6:11) 4. Computational Complexity 2: NP We introduce, and illustrate, the class NP, that consists of all “yes-no” questions for which there is a certificate for a “yes” answer whose correctness can be verified with an algorithm whose running time is polynomial in the input size. (9:04) 5. Computational Complexity 3: P = NP? Any problem that is P is also NP, but is the converse also true? This has been an open problem for decades, and is an area of active research. (1:56) 6. Revisiting Euler Circuits and Bipartite Graphs In the Euler certificate case, there is a certificate for a no answer. We introduce and illustrate examples of bipartite graphs. (8:30) 7. Dirac’s Theorem If G is a graph on n vertices, and every vertex h...

Download a PDF of the paper titled Finding Hamiltonian cycles with graph neural networks, by Filip Bosni\'c and 1 other authors Abstract: We train a small message-passing graph neural network to predict Hamiltonian cycles on Erdős-Rényi random graphs in a critical regime. It outperforms existing hand-crafted heuristics after about 2.5 hours of training on a single GPU. Our findings encourage an alternative approach to solving computationally demanding (NP-hard) problems arising in practice. Instead of devising a heuristic by hand, one can train it end-to-end using a neural network. This has several advantages. Firstly, it is relatively quick and requires little problem-specific knowledge. Secondly, the network can adjust to the distribution of training samples, improving the performance on the most relevant problem instances. The model is trained using supervised learning on artificially created problem instances; this training procedure does not use an existing solver to produce the supervised signal. Finally, the model generalizes well to larger graph sizes and retains reasonable performance even on graphs eight times the original size. arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv...

A complete Bipartite graph $K_$ has a Hamilton cycle if and only if $m= n$. Is this correct? $\begingroup$ Your proof looks good. I would justify the existence of a repeated vertex using the pigeonhole principle. You also say here: "This leads to a contradiction since a cycle cannot have repeating vertices." I would note more strongly that a Hamiltonian cycle visits each vertex exactly once. It clarifies for the reader. $\endgroup$ A complete bipartite graph $K_$ contains no Hamiltonian cycle. Thus, we get $m = n$, for all $m, n \geq 2$. Conversely, it is easy to see a Hamiltonian cycle $x_1, y_1, x_2, y_2, x_3, y_3, \dots, x_n, y_n, x_1$ for such graphs. Other direction can be prove in following way. As noted any cycle in bipartite graph will be of even length, and will alternate between the vertices of partite-sets. That means any Hamiltonian cycle in $K_$ will have equal number of elements from both the partite-sets and as it covers entire vertex set, together we get $m=n$.

• العربية • Català • Čeština • Cymraeg • Deutsch • Español • فارسی • Français • 한국어 • Bahasa Indonesia • Italiano • עברית • Lietuvių • Magyar • Nederlands • 日本語 • Norsk bokmål • Polski • Português • Română • Русский • Simple English • Slovenščina • Српски / srpski • Suomi • Svenska • தமிழ் • ไทย • Türkçe • Українська • Tiếng Việt • 中文 In the Hamiltonian path (or traceable path) is a Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian paths and cycles are named after Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the Despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Definitions [ ] A Hamiltonian path or traceable path is a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a Hamiltonian graph. Similar notions may be defined for A A Hamilton maze is a type of logic puzzle in which the goal is to find the unique Hamiltonian cycle in a given graph. Examples [ ] Any Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to Hamiltonian cycle only if its endpoints are adjacent. All Hamiltonian graphs are An G (a G exactly once. This tour corresponds to a Hamiltonian cycle in the L( G), so the line graph of every Eulerian graph is Hamiltonian. Line graphs may...