Conditions for hamiltonian cycle. , a Hamiltonian path) in G is a cycle (resp.

Conditions for hamiltonian cycle Let D be a strongly connected balanced bipartite digraph of … May 27, 2025 · Hamiltonian Cycles: Theory and Practice Hamiltonian cycles are a fundamental concept in graph theory, with far-reaching implications in various fields, including computer science, operations research, and network design. A graph that possesses a Hamiltonian path is called a traceable graph. Hamilton describing a mathematical game. If a Hamiltonian path exists whose endpoints are adjacent, then the resulting graph cycle is called a Hamiltonian cycle (or Hamiltonian cycle). Dirac saw a natural method for supplying the necessary edges, using the minimum degree d(G). Hamiltonian cycle is a Hamiltonian path that is a cycle; that is, a path that starts and ends at the same vertex and visits all vertices exactly once. Hamiltonian path implies connected and at most two nodes of degree one. Eulerian Graphs An Eulerian circuit is a cycle in a connected graph G that passes through every edge in G exactly once. In fact, the question whether a given graph has a Hamiltonian cycle is known to be NP-complete – a technical term that, for all practical purposes, implies that this question Apr 15, 2005 · A Hamiltonian cycle is a spanning cycle in a graph, i. A cycle in G is a closed trail that only repeats the rst and last vertices. Vertices P vs. For the moment, take my word on that but as the course progresses, this will make more and more sense to you. More complex examples can be constructed using various graph operations, such as adding edges or vertices to existing graphs. A Hamiltonian cycle is a closed Hamiltonian path. In general, the problem of finding a Oct 15, 2024 · For the Hamiltonian cycle problem, we will find some necessary conditions and some that are • suỶ轵cient. Graphs with a spanning cycle are called Hamiltonian Graphs. Apr 15, 2019 · We prove two sufficient conditions for Hamiltonian cycles in balanced bipartite digraphs. on Genetic and Evolutionary Computing Jun 13, 2025 · The vertex degree plays a crucial role in the existence of Hamiltonian cycles, as a vertex with a high degree is more likely to be part of a Hamiltonian cycle. What properties must a Hamil-tonian graph have? We're going to A graph meeting the conditions of Ore's theorem, and a Hamiltonian cycle in it. The problem may specify the start and end of the path, in which case the starting vertex s and ending vertex t must be identified. A Hamiltonian cycle (resp. , a cycle through every vertex, and a Hamiltonian path is a spanning path. This extension of the cycle concept allows us to avoid the undesirable additional condition 3 in almost all Hamiltonian sufficient cond A leaf of a tree is a vertex with degree one. Hamiltonian cycle: a cycle visiting every vertex exactly once and returning to start. This allows the construction of a recursive algorithm A digraph D is hamiltonian if it contains a cycle passing through all the vertices of D. We covered the basics of what defines a Hamiltonian path and a Hamiltonian cycle. 1 is Hamiltonian also a is Hamiltonian a pa path h that contains all the nodes in V(G) but does not characterization of Hamiltonian graphs exists, We begin our investigation of sufficient early results. 0 license and was authored, remixed, and/or curated by Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier via source content that was edited to the style and standards of the LibreTexts platform. In this paper, we revisit the famous Hamiltonian path problem and present new sufficient conditions for the existence of a Hamiltonian path in a graph. R. If the start and end of the path are neighbors (i. Find a Hamiltonian cycle in a graph, or explain why one does not exist. Hamiltonian Cycles graph is Hamiltonian said if to it be contains a spanning cycle is Hamiltonian called ofa G, cycle and Hamiltonian G is said(the to graph be a graph in Figure 1. Introduction Hamiltonian cycle in a graph G is a cycle spanning the vertex set of G, and a graph is said to be Hamiltonian if it contains a Hamiltonian cycle. Unfortunately, this problem is much more difficult than the corresponding Euler circuit and walk problems; there is no good characterization of graphs with Hamilton paths and cycles. Back in the world of Eulerian graphs, before we prove the big result we want, we will take a detour3 to discuss cycle decompositions. We also define a seemingly new family of bipartite graphs, which we call the Z graphs. [1] The Hamiltonian cycle Hamiltonian paths and cycles A Hamiltonian path is a path in a graph that visits each vertex exactly once. Hamiltonian Cycle is NP-complete A Hamiltonian path encodes a truth assignment for the variables (depending on which direction each chain is traversed) For there to be a Hamiltonian cycle, we have to visit every clause node We can only visit a clause if we satisfy it (by setting one of its terms to true) Nov 14, 2025 · Hamiltonian Path and Cycle: Definitions: Hamiltonian path: a path visiting every vertex exactly once. It will be helpful to remember that directed Dec 4, 2021 · Determining a graph does not contain Hamiltonian path is very difficult. . Finding Hamiltonian Cycles Hamiltonian: A cycle C of a graph G is Hamiltonian if V (C) = V (G). Hamiltonian if it has a Hamiltonian cycle. Jan 1, 2007 · The following sufficient conditions to assure the existence of a Hamiltonian cycle in a simple graph G of order n ≥ 3 are well known. Such a cycle is commonly referred to as a Hamiltonian Cycle. Unlike Euler paths, there is no simple Oct 1, 2023 · The simplest is a cycle, C n: this has only n edges but has a Hamilton cycle. It will be helpful to remember that directed cycle is a type of circuit that doesn’t repeat any edges or vertices. May 1, 2018 · 2. Today, however, the constant stream of results in this area continues to supply us with new and interesting Jan 1, 2012 · In addition, necessary and (or) sufficient conditions for existence of a Hamiltonian cycle are investigated. share a common edge), the path can be extended to a cycle called a Hamiltonian cycle. One solution is shown in the second diagram above. Beginning with Dirac’s Theorem [93] in 1952, the approach taken to developing sufficient conditions for a graph to be hamiltonian usually involved some sort of edge density condition; providing enough edges to overcome any obstructions to the existence of a hamiltonian cycle. A Hamiltonian cycle is a closed path in a graph that visits each vertex exactly once. Euler Circuit: A connected graph has an Euler circuit if and only if every vertex has an even degree. , a path) that visits all the vertices of G. 1 Overview In this lecture we discuss the Hamiltonian cycle and path problems, with an emphasis on grid graphs, and use these problems to prove some NP-hardness results for games and lawn mowing. However, many graphs fail to meet any of these conditions. It is known that the problem of deciding whether a given graph is Hamiltonian or traceable is NP-complete. In this note, we will be concerned with the degree conditions. Apr 1, 2016 · A cycle or path passing through all the vertices of a graph is called Hamiltonian. It came from a letter by W. In this paper we present two theorems stating sufficient conditions for a graph to possess Hamiltonian cycles and Hamiltonian paths. In this paper we continue to study the number of cycles in 2-factors. Hamiltonian Cycle We can construct a reduction from 3SAT to HAM Essentially, this involves coding up a Boolean expression as a graph, so that every satisfying truth assignment to the expression corresponds to a Hamiltonian circuit of the graph. Give conditions (necessary or sufficient) for a graph to be Hamiltonian. Solve the Traveling Salesman Problem for small instances. Theorem 1 (Dirac’s theorem). 4: Hamiltonian Circuits is shared under a CC BY-SA 4. They are named in honour of the great Irish mathematician, physicist, and astronomer, Sir William Rowan Hamilton (1805-1865). e. In this blog, we’ll dive into the algorithm, its practical applications, and how it addresses real-world challenges. Conf. A graph is said to be Eulerian if it contains an Eulerian Cycle, a cycle that visits every edge exactly once and starts and ends at the Nevertheless George Hendry pioneered the study of cycle extendability by proving the following statement for many types of graphs possessing conditions sufficient for Hamiltonicity: A non-Hamiltonian cycle in a graph with a property that guarantees it is Hamiltonian also guarantees it is cycle extendable. Sufficient Conditions for Hamiltonian Graphs graph G is defined to hamiltonian if it has a cycle containing all of the vertices of G. It decides if a directed or undirected graph, G, contains a Hamiltonian path, a path that visits every vertex in the graph exactly once. If a graph G has at least 3 vertices and the degree of every vertex of G is at Oct 17, 2024 · The oldest Hamiltonian cycle problem in history is finding a closed knight’s tour of the chess-board: the knight must make 64 moves to visit each square once and return to the start. For example, the following graph is 2-connected but does not have a Hamiltonian cycle (check that this is indeed the case). We need to write a function that returns 2 if the graph contains an eulerian circuit or cycle, else if the graph contains an eulerian path, returns 1, otherwise, returns 0. Dirac's theorem and Ore's theorem, discussed earlier, provide sufficient conditions for the existence of Hamiltonian cycles based on vertex degree. That’s exactly a Hamiltonian cycle in the graph we just drew. Sir William Rowan Hamilton Problem 9. There are some useful conditions that imply the existence of a Hamilton cycle or path, which typically say in some form that there are Jul 26, 2025 · Hamiltonian Cycle or Circuit in a graph G is a cycle that visits every vertex of G exactly once and returns to the starting vertex. Furthermore, in order to solve Hamiltonian cycle problems, some algorithms are Euler vs. Figure 12 7 3: Closed Walks, Circuits, and Directed Cycles The goal of Hamilton's puzzle was to find a route along the edges of the dodecahedron May 27, 2025 · The above graph is a simple cycle graph C 4 C 4, which is also a Hamiltonian cycle. This reduction is much more intricate than the one for IND. Over the last seventy years, the following problem has received a great deal of attention: under what conditions does a graph G with a Hamiltonian cycle contain another long cycle distinct from ? Of course, for this question to be This page titled 6. A Hamiltonian cycle on the regular dodecahedron. 1. 1 Hamiltonian Cycles Recall that a spanning cycle is cycle in some graph that contains all vertices of that graph. We will not try to solve the 8 × 8 problem today. We're going to go a bit more in-depth with this concept. For general digraphs there are many sufficient conditions for existence of hamiltonian cycles in digraphs (see, e. ph is Hamiltonian, if it contains a Hamilton cycle - a simple spanning cycle. Dirac’s Theorem Recall that a Hamiltonian cycle in a graph G = (V, E) is a cycle that visits each vertex exactly once. the vertices proceds clockwise, and we label the colours with Conclusion In this chapter, we explained the concept of Hamiltonian paths in discrete mathematics. 657n), so exponential The problem: Suppose that P knows a Hamiltonian Cycle for a graph G. We do know of some necessary conditions (any graph that fails to meet these conditions cannot have a Hamilton cycle) and some sufficient conditions (any graph that meets these must have a Hamilton cycle). On the other hand, Figure 5 3 1 shows graphs with just a few more edges than the cycle on the same number of vertices, but without Hamilton cycles. 1 Dirac [4] If δ (G) ≥ n / 2 , then G is Hamiltonian. A tree with at most leaves is called an tree, where ≥ 2 is an integer. Definition 2. Hamilton Edges vs. Unlike for Euler cycles, no simple characterization of graphs with Hamiltonian cycles is known. Basic Necessary Conditions: Graph must be connected (for undirected); strongly connected usually required for directed cycle. Nov 24, 2019 · Hamiltonian cycle implies biconnected, which in turn implies that every node has degree at least two. See for example this particular graph. Definition and Examples of Hamiltonian Cycles A Hamiltonian Cycle is a closed path in a Jul 12, 2021 · This is a hard problem in general. A cycle of G containing every vertex of G is called hamiltonian cycle of G. Hamiltonian-Cycle: In the undirected graph G = (V, E), hamiltonian cycle is a simple cycle that contains each vertex in V (starts from one vertex, travels every vertex, then returns to the first one). We construct a simple algorithm for generating Hamiltonian cycles on two-dimensional square lattices. Aug 13, 2025 · A complete guide to Hamiltonian graphs, covering path and cycle concepts with real-world applications and how to determine one using code with examples. Hamiltonian path A Hamiltonian cycle around a network of six vertices Examples of Hamiltonian cycles on a square grid graph 8x8 In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. Consider a graph with \ ( 64 \) vertices in an \ ( 8 \times 8 \) grid, with each vertex corresponding to a square May 27, 2025 · Introduction to Hamiltonian Cycles Hamiltonian Cycles are a fundamental concept in Discrete Mathematics, named after the mathematician William Rowan Hamilton. Hamiltonian Paths and Cycles (2) Remark In contrast to the situation with Euler circuits and Euler trails, there does not appear to be an efficient algorithm to determine whether a graph has a Hamiltonian cycle (or a Hamiltonian path). The significance of the theorems is discussed, and it is shown that the famous Ore's theorem directly follows from our result. 157. However, we will think about the problem for Hamiltonian Cycles and Paths. No vertex cut of size k can split graph into more than k components for a Basic Definition : Hamiltonian Cycle: If G = (V, E) is a graph or multi-graph with |V|>=3, we say that G has a Hamiltonian cycle if there is a cycle in G that contains every vertex in V exactly once . How can she prove this to V in zero-knowledge? Before we look at the solution to Hamilton's puzzle, let’s review some vocabulary we used in Figure 12. Named for Sir William Rowan Hamilton, this problem traces its origins to the 1850s. Figure 5 3 1: A graph with a Hamilton path but not a Hamilton cycle, and one with neither. 1 Introduction A graph G is hamiltonian if it contains a spanning cycle. We discussed their unique conditions and challenges, and saw examples of when these paths and cycles appear in different graph types. The first Mar 12, 2013 · 1 I have knowledge of the necessary and sufficient condition for an undirected graph contains a Hamiltonian cycle and an Eulerian circuit, but is there a necessary and sufficient condition for directed graphs? Nov 23, 2024 · Hamiltonian cycles are crucial in solving optimization problems, especially in logistics, network design, and game theory. A Hamiltonian path is a traversal of a (finite) graph that touches each vertex exactly once. Necessary and Sufficient Conditions: Condition for Hamiltonian based on Linear Diophantine Equation Systems with Cycle Vector, 2009 3rd Intl. Let G be a graph. Feb 3, 2025 · Conditions for Euler Paths and Circuits Euler Path: A connected graph has an Euler path if and only if it has exactly zero or two vertices of odd degree. However, there does exist theorems which give sufficient conditions for the existence of a Hamiltonian cycle. There are two vertices with degree less than n /2 in the center of the drawing, so the conditions for Dirac's theorem are not met. Definition Sufficient conditions for Hamiltonian cycles are specific criteria that, when met, guarantee the existence of a Hamiltonian cycle in a graph. Properties of Hamiltonian Cycles Conditions for a Graph to Have a Hamiltonian Cycle While there is no known necessary and sufficient condition for a graph to have a Hamiltonian cycle Jun 8, 2025 · Given an undirected connected graph with v nodes, and e edges, with adjacency list adj. This sufficient condition for Hamiltonian path found in this link ClickHere does not The Hamiltonian path problem is a topic discussed in the fields of complexity theory and graph theory. Note that if a graph has a Hamilton cycle then it also has a Hamilton path. A graph is Hamiltonian if it has a closed walk that uses every vertex exactly once; such a path is called a Hamiltonian cycle 1 Overview In this lecture we discuss the Hamiltonian cycle and path problems, with an emphasis on grid graphs, and use these problems to prove some NP-hardness results for games and lawn mowing. In this article, we will explore the definition, properties, and applications of Hamiltonian Cycles, as well as their importance in various fields. , a Hamiltonian path) in G is a cycle (resp. Ore's theorem is a result in graph theory proved in Aug 26, 2020 · Hamiltonian Paths and Cycles Sir William Rowan Hamilton was an Irish mathematician and the inventor of icosian calculus — which he used to investigate closed edge paths on a dodecahedron that What are Hamiltonian cycles, graphs, and paths? Also sometimes called Hamilton cycles, Hamilton graphs, and Hamilton paths, we’ll be going over all of these topics in today’s video graph A graph G is Hamiltonian if it has a Hamiltonian cycle, that is, if the cycle Cn is a spanning subgraph of G. In section 6, the algorithm CHAR-VEC applied on the subdivision of G gives a necessary and su cient condition, based on g; for a graph to be Hamiltonian. If a graph contains a Hamiltonian cycle, it is called Hamiltonian graph otherwise it is non-Hamiltonian. A graph G, containing a Hamiltonian cycle or path, is called Hamiltonian or traceable correspondingly. Hamiltonian Paths Hamiltonian Path: A path in a graph that visits every vertex exactly once. Jan 2, 2025 · Hamilton’s Puzzle Before we look at the solution to Hamilton's puzzle, let’s review some vocabulary we used in Figure 12 7 3 . We show that our sufficient condition can also ensure Hamiltonicity unless the graph is a Z graph. Not all graphs have a spanning cycle. Because of topological properties of the interior bounded by a Hamiltonian cycle, the problem of enumerating Hamiltonian cycles reduces to coloring simply-connected regions of square grids, subject to simple, locally-implemented rules. The hamiltonian problem is generally considered to be determining conditions under which a graph contains a spanning cycle. Formal Language: HAM-CYCLE = {<G>: G is a hamiltonian graph} Hamiltonian graphs and TSP Hamiltonian path (named for William Rowen Hamilton, 1805-1865) is a path that visits every vertex in a graph exactly once. These conditions help in identifying whether a graph contains a cycle that visits each vertex exactly once before returning to the starting vertex, a crucial concept in combinatorial optimization and graph theory. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where 1Uk U jV–Gƒj 4 . A Hamiltonian path in a graph is a path involving all the vertices of the graph. However,graphtheoryknowsnosimplenecessary and suỶ轵cientcondition. 23. As for (closed) Eulerian trails, we are interested in the question of whether a given graph has a Hamiltonian cycle/path. Theorem 1. , [5], [7], [10], [15], [16], [21]). g. However, these two vertices are adjacent, and all other pairs of vertices have total degree at least seven, the number of vertices. A Hamiltonian path, also called a Hamilton path, is a graph path between two vertices of a graph that visits each vertex exactly once. Objectives Define Hamiltonian cycles and graphs. NP No necessary and sufficient conditions for a Hamiltonian cycle No good algorithm for finding one (there are known algorithms with running time O(n22n) and O(1. Nov 5, 2013 · In this paper, we present Dirac-type sufficient conditions for a bipartite graph to possess a Hamiltonian path. 3. udxztyo paw wihxx cbsd rphqjj yjy ygnp jbplixfm rmzuicpx orkb ayudxkr wnza uhqwmc duumo jhpnq