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When you’ve engaged with graphs in computer science, you may have encountered the expression Hamiltonian graph. In simple terms, a Hamiltonian graph is one that features a Hamiltonian path. This path traverses each vertex exactly one time. If this route starts and finishes at the same vertex, it forms a Hamiltonian cycle. Such graphs are not merely theoretical; they possess practical applications like optimizing delivery paths or arranging genomes. In this blog post, you will discover what a Hamiltonian graph is, along with important characteristics and uses of Hamiltonian graphs, and using straightforward code in C++, you will be capable of checking if your graph is Hamiltonian or not.

Table of Contents:

• What Defines a Hamiltonian Graph
• Essential Properties of Hamiltonian Graphs
• How to Ascertain if a Graph is Hamiltonian
• Illustration of a Hamiltonian Graph
• Distinctions Between Hamiltonian and Eulerian Graphs
• Real-World Uses of Hamiltonian Graphs
• Conclusion

What Defines a Hamiltonian Graph

A graph must possess a Hamiltonian path to qualify as a Hamiltonian graph. Imagine a scenario where you are traveling to a city and want to visit every location (node) without missing any and without revisiting. If such a pathway is available, it is deemed a Hamiltonian path. Now, if you are able to traverse all locations exactly once and return to your starting point, you have discovered a Hamiltonian cycle. It’s crucial to understand that not all edges need to be included in the cycle; as long as at least one Hamiltonian cycle exists, the graph is classified as Hamiltonian.

A Hamiltonian graph is essentially a graph featuring such a cycle. It links all nodes in such a manner that you can commence at any vertex, visit all other vertices once, and then return to the origin.

For a specific example, consider a graph with 5 vertices, and you can trace a route like 0→1→2→4→3→0, which indicates that it has a Hamiltonian cycle, thus the graph is Hamiltonian.

These graphs are intriguing because figuring out whether a graph is Hamiltonian is not as straightforward as confirming its connectedness. Instead, there is no universal guideline, and the process can become complex swiftly. This illustrates why understanding their structure is vital for solving related problems efficiently.

Essential Properties of Hamiltonian Graphs

When you opt to work with Hamiltonian graphs, several characteristics will assist you in determining if a path within the graph genuinely constitutes a Hamiltonian cycle. There is certainly no singular rule for this, yet a handful of theorems offer valuable insights.

1. Dirac’s Theorem

Assuming a graph comprises N vertices in total, where N is at least 3, and each of these N vertices connects to a minimum of N/2 other vertices, the graph is guaranteed to be Hamiltonian.

Example:

In a graph consisting of 6 vertices, each vertex must link with at least 3 others. If this condition holds for every vertex, there is a likelihood that the graph is Hamiltonian.

2. Ore’s Theorem

If for every pair of unconnected vertices u and v, the sum of their degrees (defined as the number of edges adjacent to each vertex) is at least equal to N, the graph is Hamiltonian.

Example:

In a graph where N = 5, if two nodes that lack connections have degrees summing up to 5 or higher, this is a strong sign of a Hamiltonian graph.

3. NP-Completeness

Determining the existence of a Hamiltonian path is classified as an NP-complete problem, meaning it is relatively simple to verify a solution’s correctness but exceedingly challenging to pinpoint a solution for extensive graphs.

4. Every Complete Graph is Hamiltonian

When every vertex connects to each other, it forms a complete graph, and such a graph always contains a Hamiltonian cycle.

5. Not All Graphs with Hamiltonian Paths are Hamiltonian Graphs

It is not universally true that a graph containing a Hamiltonian path (visiting each vertex at least once) automatically includes a cycle. To qualify as a Hamiltonian graph, it must also offer a route back to the starting location.

A Graph and Its Hamiltonian Cycle

In the image below, you can observe the Hamiltonian cycle outlined by red lines. Notice how all vertices are visited once, and an edge from C to E is excluded, yet it still constitutes a Hamiltonian graph.

How to Ascertain if a Graph is Hamiltonian

If you wish to ascertain whether a graph is Hamiltonian, what you are effectively inquiring is whether this graph possesses a cycle that visits all vertices in a single traversal (i.e., each vertex only once) and returns to the initial vertex. Unfortunately, no overarching guideline exists for this. This is categorized as an NP-complete problem, i.e., it becomes increasingly challenging to solve as the graph expands. We will attempt to utilize the methods we outlined earlier to demonstrate how to apply them:

1. Apply Dirac’s Theorem

The theory states: If a graph contains n ≥ 3 vertices, and each vertex has a degree ≥ n/2, then the graph is Hamiltonian.

Example: If your graph features 6 nodes, and each node connects to at least 3 others, you’re in good shape.

2. Implement Ore’s Theorem

Ore’s theorem asserts:

If the degree sums of any two unconnected vertices are ≥ n, then the graph is Hamiltonian.

Both theorems provide sufficient, but not essential conditions. Thus, even if your graph does not satisfy them, it could still be Hamiltonian.

3. Brute-Force Method

This method can be utilized for small graphs. Here you may attempt all conceivable configurations…

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Permutations of various nodes are examined to determine if any create the desired Hamiltonian cycle. This technique is undoubtedly slow, yet it proves effective for smaller graphs.

C++ Sample Code:

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Output:

Clarification: The aforementioned C++ program assesses whether a specified graph contains a Hamiltonian cycle using a brute-force approach. It generates a list of indices for all vertices and forms every possible permutation of those indices. During each permutation, it inspects every subsequent pair of vertices along the path to verify their adjacency (using the adjacency matrix) and checks if the last vertex connects back to the first, thereby completing the cycle. Upon discovering even one such valid cycle, it returns true; if it exhausts all options, it returns false. This straightforward and accurate algorithm is suitable for small graphs, but becomes inefficient for larger datasets due to its factorial time complexity.

Example of a Hamiltonian Graph

For example, consider a graph comprising 5 points (or vertices), labeled from 0 to 4. The goal of the problem is to find a path that visits each of these points exactly once (and returns to the starting point), thereby forming a Hamiltonian cycle. Below is a C++ representation of an adjacency matrix, showcasing how each vertex correlates with one another:

Backtracking Strategy using C++:

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Output:

Explanation: Each line within the matrix depicts the connected vertices. Here, 0→1→ 2 →4 →3 →0 represents the path where each vertex is visited only once before returning to the start. This sequence forms a Hamiltonian cycle. The provided code exemplifies a fundamental case of how a Hamiltonian graph is utilized in practice.

Differences Between Hamiltonian and Eulerian Graphs
While Hamiltonian and Eulerian graphs may appear similar, they serve distinct purposes in graph theory. Understanding their differences is critical for applying the correct method when tackling a path/cycle-related problem.

Feature	Hamiltonian Graph	Eulerian Graph
Definition	Has a cycle (or path) that visits every vertex exactly once	Contains a cycle (or path) that traverses every edge exactly once
Visits	Each vertex one time	Every edge one time
Repetition	Edges may repeat, but vertices cannot	Vertices can be revisited, but every edge is used only once
Requirements	Must meet Hamiltonian cycle/path conditions (for example, Dirac’s or Ore’s theorem)	All vertices must have even degree (for an Eulerian cycle)
Complexity	NP-complete (more challenging to solve)	Polynomial time (easier to verify)

Imagine a Hamiltonian graph as a delivery route where you must stop at each house just once. In contrast, an Eulerian graph resembles a street-cleaning route, where no street (edge) can be cleaned twice, even if you might pass by the same house more than once.

Real-World Applications of Hamiltonian Graphs

Hamiltonian graphs are not merely theoretical concepts found in textbooks; they also address tangible problems across various fields. Here are some real-world scenarios:

1. Traveling Salesman Problem (TSP)

Assume you are a delivery person aiming to visit each city on your list and return home along the shortest route. This is exactly what the TSP addresses. A Hamiltonian cycle guides you to the shortest circuit with no repeated elements, making it ideal for route optimization.

2. Genomic Sequencing

In molecular biology, sequencing DNA fragments (termed reads) entails reconstructing the original sequence, often utilizing Hamiltonian paths. Each read is treated as a node, and they are organized in such a way to recreate the original DNA sequence using a Hamiltonian path.

3. Optimal Path Planning

In fields such as urban planning, robotics, and logistics, it is essential to identify routes that cover all necessary checkpoints without looping back. Hamiltonian graphs assist in determining if such a path is feasible and outline the necessary steps.

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should be to attain it.

4. Computer graphics and rendering

Hamiltonian cycles are advantageous in several graphics systems, particularly in mesh rendering or wireframe modeling, to determine how to traverse each edge or vertex with a single drawing operation.

5. Circuit Design

Hamiltonian paths also aid in electronic circuit design by facilitating the selection of efficient component arrangements with minimal wire usage and decreased overlap. (by attempting to use the shortest path to reduce wire length or to avoid overlap among pairs of points, or in circumventing unnecessary jumps).

Conclusion

Understanding Hamiltonian graphs is pertinent when you are new to graph theory or tackling real-world applications, such as routing or genome sequencing, or crafting a networking system. In a Hamiltonian graph, a Hamiltonian path (or Hamiltonian cycle) traverses each vertex of the graph once and only once, and the presence of such paths can expedite problem-solving. You have also uncovered significant theorems, such as those of Dirac and Ore, and recognized the differences between Hamiltonian graphs and Eulerian graphs, as well as implemented a program to verify Hamiltonian cycles. With this foundation, you may now explore more complex applications or transition to related graph challenges.

Hamiltonian Graph- FAQs

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